Absolutely YES

"retaliating with a nuclear weapon" implies that they not only had a nuke, but had the means to deliver it to a meaningful target, which means continental USA.
A reliable means to deliver the bomb, and a heavy bomber capable of doing so, and conquered airfields within range of continental USA, and sufficient air parity or supremacy to reasonably guarantee the bomber would get through. All of the items in bold are required, and none of them were available historically.

In other words, the assumption that Japan has a deliverable nuclear bomb, carries with it the perquisite that they are doing massively better in the war than they historically did. In that scenario, the US public would most certainly have been much more likely to consider peace talks, rather than pursuing Japan's annihilation.

If Japan had had the bomb in 1945, their obvious use would have been tactical and not strategical. As other answers point, Japan didn't have the ability to deliver the bomb to American soil, but it had the ability to drop the bomb on American troops. A nuke could even have been used as a large explosive trap in prospective landing grounds in Japan.

Therefore, a Japanese nuke would have prevented conventional effort from the US - that it, it could have prevented the invasion of Japan planned for 1946. That would have left nuclear strategic bombing of Japan as the only alternative.

Furthermore, Japan having the bomb would give an additional reason to the allies to want unconditional surrender of Japan. Any armistice that didn't strip the nuke from Japan, would give Japan the opportunity to get the ability to deliver it to US soil.

The key question is if each side knows about the other's bombs (or bomb program, same effect).

If they believed they were alone, they would likely use it as we did - to best effort to end the war in their favor. This could result in a rude surprise if the other side did in fact have them.

The more likely case is they would know. By developing a bomb, they'd learn to spot the subtle signs of a society developing a bomb. "Ah, we have this sprawling complex on Hokkaido which has very particular trucks going in and out (notably, not coal trucks) and very particular scientists... suddenly we realize what the sprawling complex at Hanford is." Knowing what to look for, they'd uncover a lot more stuff... for instance they would have already contemplated "where would the US test such a weapon?" and have spies already looking when the Trinity test went off. An hour later, a neutral intermediary delivers a message from the Japanese: "Welcome to the nuclear age."

Once there is mutual realization (or presumption of same), fear of the weapon will send the diplomatic corps into overdrive. It's likely that both will drive for a consensus to not use the weapons in this war, and both sides would labor to add more and more conditions, and soon this would bust out into bona-fide negotiations which may end the war.

Keep in mind, detonating a nuke on your own soil is fair play. If an enemy army is there, too bad, they're trespassing. (That is the real motivation behind North Korea's nukes; it forecloses any land-based invasion, securing their borders for good.) It would do the same for Japan, removing a land invasion from the table, which was a critical priority for the nation.

A nuclear Imperial Japan would have been wisest to preserve their nukes for deterrent purposes, and make their intentions clear through diplomacy.

Also fair game is safe, peaceful detonations on your own territory. The Soviets dug canals with nukes, and no nation considered that a provocation. Imagine if instead of fighting to the death over small islands like Iwo Jima, their garrison did controlled detonations to reshape or contaminate the island so it could not be used by the enemy (i.e. as an air base).

Although all of the above answers are good, there is a difference between delivering a single bomb to a city, which only takes a single plane (or submarine). And Japan had built three submarine carriers which were designed to launch planes capable of carrying 800kg bombs, while they also built 47 submarines capable of launching a single plane, such as the Yokosuka E14Y. These aircraft were not able to deliver a large enough bomb. But since we are considering alternate history, the real question is whether Japan could build a plane (that the I-400 could carry) which could lift their conjectured nuclear device. And given enough research, could a smaller bomb be built? An alternative would be to deliver a nuclear device to a port city via submarine, and use a suicide attack to detonate the device in the port.

And would the U.S. or Japan be deterred by the threat of a nuclear device? That would be a matter of opinion, or speculation. But evidence shows that Japan was not deterred by massive bombing of their home islands. And the U.S. had not been deterred from response by the substantial setback from Pearl Harbor. Would the U.S. have been deterred had San Francisco or Los Angeles been heavily damaged by a nuclear blast, or would the U.S. have been enraged? My guess would be the latter; but again, speculation.

The simple answer is no. The historical fallacy at the heart of this question is that Japan surrendered because the USA dropped two nuclear bombs on Japanese cities.

It is considered that the real reason for Japan's surrender was the declaration of war against Japan by the Soviet Union. Japan was well aware that Russia had a score to settle with them over the Russian defeat by the Japanese in the early 1900's. Japan was driven close to defeat fighting the Allies in the Pacific, so the opening up of a 'Western Front' fighting Russian forces would have led to inevitable defeat.

The majority of Japanese cities had already been destroyed by American bombing. In fact, it was difficult for the Americans to select target cities for nuclear bombing. They wanted to make it clear that nuclear weapons had been solely responsible for destroying the targets. previous bombing raids on a city might have left behind, for example, time bombs to annihilate the target.

There is also the distinct possibility that the use of the atomic bombs were used to demonstrate to the Soviet Union that the USA possessed a super-weapon. This was intended to shape global politics in the post-war era.

If Japan had its nuclear weapons it would have been trapped between two massively powerful adversaries. Namely, the USA and the USSR. Even the USA had a limited number of atomic bombs. Often the estimate is three. These would have been used up with the Trinity test and the bombs used on Hiroshima and Nagasaki. In the OP's alternative history scenario the USA could have easily had many more. (Indeed, it is conceptually possible that any alternative history where there are more nuclear weapons at the end of the Second World War the USSR could also have its own nuclear weapons too.)

Decision makers in Japan would be faced with determining how they could use their nuclear weapons most effectively. This means against the USA and the USSR. Even if only they are confronted with a nuclear-armed USA, they will still have to deal with Soviet conventional forces on a grand scale. Their best option seems to be surrender before they are engaged in fighting Soviet forces. This is essentially similar to the sequence of events in history as we know it.

What might be different is that if there are more nuclear weapons, is that more Japanese cities will be destroyed with nuclear weapons before Japan surrendered.

Please note this answer is based on historical research that demonstrates the Japanese surrender was more due to the Soviet declaration of war than the historical myth concerning the nuclear bombing of Hiroshima and Nagasaki.

ADDENDUM:

@sphennings in comments requested links concerning the historical background. Admittedly the real historical research will be in books and history journals there are links to various aspects of the Japanese surrender.

From the Wikipedia entry on the Soviet-Japanese War (1945) illustrates the complexity of factors involved. To assume the surrender of Japan in 1945 was caused by the two atomic bombing is too simplistic. Simplification on this scale is the stuff of myth.

From the time of the first major Japanese military defeats in the Pacific in the summer of 1942, the non-military leaders of Japan had come to realise that the Japanese military campaign was economically unsustainable — as Japan did not have the industrial capacity to simultaneously fight the United States, China and the British Commonwealth and Empire — and there were a number of initiatives to negotiate a cessation of hostilities and the consolidation of Japanese territorial and economic gains. Hence, elements of the non-military leadership had first made the decision to surrender as early as 1943; the major issue was the terms and conditions of surrender, not the issue of surrender itself. For a variety of diverse reasons, none of the initiatives were successful, the two major reasons being the Soviet Union's deception and delaying tactics, and the attitudes of the "Big Six", the powerful Japanese military leaders.[26] (Refer to Surrender of Japan for more detail.)

The Manchurian Strategic Offensive Operation, along with the atomic bombings of Hiroshima and Nagasaki, combined to break the Japanese political deadlock and force the Japanese leaders to accept the terms of surrender demanded by allies.

In the "Sixty years after Hiroshima" issue of the Weekly Standard, American historian Richard B. Frank points out that there are a number of schools of thought with varying opinions of what caused the Japanese to surrender. He describes what he calls the "traditionalist" view, which asserts that the Japanese surrendered because the Americans dropped the atomic bombs. He goes on to summarise other points of view.[27]

Tsuyoshi Hasegawa's research has led him to conclude that the atomic bombings were not the principal reason for Japan's capitulation. He argues that Japan's leaders were impacted more by the swift and devastating Soviet victories on the mainland in the week following Joseph Stalin's August 8 declaration of war because the Japanese strategy to protect the home islands was designed to fend off a US invasion from the South, and left virtually no spare troops to counter a Soviet threat from the North. This, according to Hasegawa, amounted to a "strategic bankruptcy" for the Japanese and forced their message of surrender on August 15, 1945.[28][29] Others with similar views include The "Battlefield" series documentary,[20][21] among others, though all, including Hasegawa, state that the surrender was not due to any single factor or single event.

A broader picture of the events leading to Japan's surrender can be found in the Wikipedia entry on the Surrender of Japan.

Foreignpolicy.com announced that "The Bomb Didn't Beat Japan ... Stalin Did"

The U.S. use of nuclear weapons against Japan during World War II has long been a subject of emotional debate. Initially, few questioned President Truman’s decision to drop two atomic bombs, on Hiroshima and Nagasaki. But, in 1965, historian Gar Alperovitz argued that, although the bombs did force an immediate end to the war, Japan’s leaders had wanted to surrender anyway and likely would have done so before the American invasion planned for Nov. 1. Their use was, therefore, unnecessary. Obviously, if the bombings weren’t necessary to win the war, then bombing Hiroshima and Nagasaki was wrong. In the 48 years since, many others have joined the fray: some echoing Alperovitz and denouncing the bombings, others rejoining hotly that the bombings were moral, necessary, and life-saving.

The following lecture on the Carnegie Council website further collaborated the historical role of the Soviet Union in Japan's surrender.

Look at the facts. The United States bombed 68 cities in the summer of 1945. If you graph the number of people killed in all 68 of those attacks, you imagine that Hiroshima is off the charts, because that’s the way it’s usually presented. In fact, Hiroshima is second. Tokyo, a conventional attack, is first in the number killed. If you graph the number of square miles destroyed, Hiroshima is sixth. If you graph the percentage of the city destroyed, Hiroshima is 17th.

Clearly, in terms of the end result—I’m not talking about the means, but in terms of the outcome of the attack—Hiroshima was not exceptional. It was not outside the parameters of attacks that had been going on all summer long. Hiroshima was not militarily decisive.

The Soviet Union’s declaration of war, on the other hand, fundamentally altered the strategic situation. Adding another great power to the war created insoluble military problems for Japan’s leaders. It might be possible to fight against one great power attacking from one direction, but anyone could see that Japan couldn’t defend against two great powers attacking from two different directions at once.

The Soviet declaration of war was decisive; Hiroshima was not.

After Hiroshima, soldiers were still dug in in the beaches. They were still ready to fight. They wanted to fight. There was one fewer city behind them, but they had been losing cities all summer long, at the rate of one every other day, on average. Hiroshima was not a decisive military event. The Soviet entry into the war was.

Even Foxnews announced on 14 August 2010 that the Soviet Offensive was the key to the Japanese surrender was eclipsed by the A-bombs.

It was a momentous turn on the Pacific battleground of World War II, yet one that would be largely eclipsed in the history books by the atomic bombs dropped on Hiroshima and Nagasaki in the same week 65 years ago. But in recent years some historians have argued that the Soviet action served as effectively as — or possibly more than — the A-bombs in ending the war.

Essentially this information can be readily found by a Google search using the search terms "surrender Japan Soviet Union".

By the late-war period (mid-1944 onward) when Japan could reasonably be expected to have an atom bomb, Japan had no way to reliably deliver such a bomb to a target in the United States.

Enola Gay and Bockscar were able to drop their bombs unhindered because Japan was desperately short on fuel, ammunition, and aircraft. Japanese policy was to ignore single aircraft flying over the home islands, reserving their limited anti-aircraft capabilities for the bomber raids (and even those didn't see much opposition).

The situation in the United States was exactly the opposite. Fighters and anti-aircraft guns were available in abundance, radar gunlaying meant those guns were highly effective, and widespread radar facilities meant that even single planes could be spotted and fighters vectored for an intercept long before the incoming planes became a threat.

In actual history, Japan was not considered a serious threat to the American mainland, so coastal defenses were spotty at best. If, on the other hand, Japan were seriously believed to have a working atom bomb and a means to deliver it, some of that abundance of defenses would be diverted to homeland guard duty. Japanese possession of a bomb would not deter American use of their own nuclear weapons, because unlike Japan, American military leaders would be confident in their ability to intercept and destroy any attack.

No this would not have deterred the U.S.

• For the concept of Mutually Assured Destruction to work, there has to have been a demonstration in a real city as to the effects.
• The United States is bigger than Japan, and we had more bases in different places. It would not have been too difficult at this point in the world to run an air blockade on Japan. As other posters pointed out, they lacked the resources for delivery.
• If they did have the capacity, they simply would have done it, rather than threatening it. It took TWO bombs for Japan to surrender, because they did not believe we could have built more. The U.S. after the Japan dropped the first one on us, would have simply bombed Japan into oblivion, both with conventional bombs and with whatever nukes we had. We just would have targeted all the airfields, and all the major cities, and just poured it all on.
• Expanding on mutually assured destruction--despite our demos, the people of the United States would be unlikely to accept a surrender, because they wouldn't understand. Only a chosen few understood, because this was top secret. Even if it wasn't it would have felt abstract. And military minds think in terms of acceptable losses. We had much more land, people and many more cities that would be difficult for them to get to. Japan 1940: 73,075,071 people. U.S. 132.1 million people.

Logistically it was much easier for the US to get to all of the Japanese cities than it was for them to get all of ours (because we are so large). I can definitely see us launching everything we had after--if there are no aircraft carriers left, they can't bomb us.

No, and we thought they were (kinda) close to one

The Germans probably could have made a bomb if they hadn't gotten caught up on the whole "heavy water" thing. In fact, Einstein warning FDR about German bomb efforts in his letter was part of what prompted the Manhattan project. Late in the war, the Germans tried to send the Japanese the Uranium they had, seeing as Japan was also working on a bomb, though they were still a ways off. However, the US captured the ship in May 1945 (and probably used the captured Uranium in our own bombs). Still, Japan getting a bomb in some alternate timeline isn't beyond the realm of plausibility.

However, as was pointed out previously, they had few good delivery methods. They had balloons to carry firebombs to the continental US from Japan itself, but these were incredibly imprecise and ineffective. A nuke would need to actually hit a city or base to be effective, so this wouldn't work (also, most fell down in the ocean before arriving). In an alternative history scenario one thing they could have done is they were working on a submersible aircraft carrier. These were actually able to bomb the US briefly (for the only time in the war). One of them slipping past US patrols to drop a bomb on San Francisco is a risky but plausible gambit. It was actually planned to use these to drop the black plague on US cities, though it never came to pass.

As for whether it would deter the US dropping its bomb, however, it absolutely would not. The Japanese were so desperate at that point in the war, that they would have dropped it immediately, so US planning would not have been effective. Plus, all the above delivery methods have risks, so if Japan warned the US to make peace "or else", the US could have prepared to intercept the bomb, and would be ready to risk it. Nuclear weapons at the time, while incredibly destructive, weren't quite as insane as they are now. A bomb dropped on California would have killed maybe 100-150 thousand people. Insane losses, sure, but close to what the US was estimating for military losses, should we have to invade the Japanese home islands conventionally. While it would certainly have provoked a response from the US, it wasn't the same as the Mutually Assured Destruction seen during the Cold War.

The US considered demonstrating the devastating effect of an atomic bomb before attacking Japan in an attempt to convince the Japanese to surrender without massive civilian casualties. They concluded that they could propose no technical demonstration likely to bring an end to the war.

For the same reasons that the US didn't announce the existence of nuclear weapons before bombing Japan, if Japan had the capability to drop an atomic bomb on the US during WWII they would have done so without warning. The most likely outcome is whichever country developed the bomb first would launch a strike at the first opportunity.

When the US dropped atomic bombs on Hiroshima and Nagasaki they had the capability to conventionally bomb Japanese cities and were doing so regularly. Japan didn't have the capacity to fly bombers over the US. The closest thing they had were high altitude balloons armed with firebombs. This delivery method wouldn't be suitable for delivering a nuclear weapon to US soil.

The US was able to launch its attack on Japan from Tinian about 1500 miles from Japan. While the Japanese did have a foothold on American soil in the Aleutian Islands these islands are much closer to Russia than mainland Alaska, let alone the rest of the US. Given the US's superior position late in the war I doubt knowledge of a Japanese atomic bomb would have discouraged the US from bombing Japan.

The US had the capability to not only build a nuke, but get and drop it on Japan, whereas by 1944 Japan had no such capacity. Thus, not many worries that Japan might nuke us.

Also, if we knew where they were developing the bomb, that would have been our prime target. Even if it were in Manchukuo, we'd have found a way to bomb it, no matter the expense, because... the threat of being nuked would have made us work even harder to (1) stop them, and (2) work faster at our own bomb.