First atomic-powered transportation in science fiction

I don't know about transportation, but if you are talking about nuclear energy as an energy source in general, then yes, sort of.

In 1934, German writer Hans Dominik wrote "Atomgewicht 500", in which a scientist creates a (fictional) synthetic element with an atomic weight of, yes, five hundred, that is used to provide energy by running turbines. This is not quite how things works in real life (in the story, the element is dropped as a pellet into a water filled boiler where it heats the water, but stops releasing energy when the working pressure of the boiler is exceeded. Wish it were that simple.), but it is a steam engine run with nuclear power.

You don't seem quite satisfied with the existing answers, maybe because the stories don't provide details of how atomic energy is used to propel the ship.

Robert A. Heinlein's Rocket Ship Galileo from 1947 describes it in some detail, as a group of young men and a nuclear physicist spend some time building the drive system of the Galileo.

The drive uses thorium to "boil" zinc to steam. The zinc steam is ejected through jets into a rocket engine bell. It provides thrust much like a chemical rocket engine, but with a higher velocity. The higher exhaust velocity combined with the higher mass of the exhaust gives the Galileo a better thrust to mass ratio than any chemically propelled rocket.

It certainly isn't the first, but it is more detailed than what I've seen in other stories.

In 1927 Karel Čapek wrote THE ABSOLUTE AT LARGE. The inventor has created a device called the "Karburator" which allows the "complete utilization of atomic energy." Using a special filament it breaks atoms into their component parts and harnesses the electrons to create electrical current. In the prototype, the current is used to move a piston that rotates a fly-wheel.

Unfortunately it has a rather serious side-effect.

The 1914 novel The World Set Free by H.G. Wells is written in the style of a futuristic history book, and it features atomic vehicles, and goes into some detail on the theory of atomic power. The novel is in the public domain and can be read here.

In Chapter 1, Section 8, which takes place before the harnessing of atomic power, one professor describes the theory that all matter emits radiation the same way radium does, and if that can be sped up, it would be a great power source.

Chapter 1 Section 1 summarizes the first experiment in which this was accomplished when a scientist named Holsten forced bismuth to rapidly radiate. It release a dangerous amount of power in an explosion that lasts 7 days. Along the way, it transmuted to different elements in phases, and ended as gold.

Chapter 1 Section 3 describes early atomic vehicles, with some technobabble describing different engines. Some engines run on bismuth and others run on lead. As a whole, atomic engines are very light for the amount of power they emit.

Chapter 2 Sections 3-4 describe atomic bombs in a way that shines a little bit of light on how atomic power works in this setting. There are some synthetic elements, the most powerful of which is Carolinum, which begins radiating when it makes contact with air. There's a quote that says that all radio-active substances decay following a half-life, and they never completely run out of energy.

In summary, this gives a detailed description of the theory of atomic power. Rereading your question, I think you might be asking about how that power is turned into motion. I can't find anything that explicitly describes that, but there is a lot of discussion of heat and explosion, and a lot of comparison to steam and coal. I think it's safe to say that H.G. Wells imagined using this to turn a turbine. I don't think he imagined the atomic energy boiling water and using that to turn an a turbine, because he says that electricity replaced the steam engine for traction in the 1930s, and atomic automobiles became practical in the 1950s.