If a clause is a direct object, its pronoun is nominative because the whole clause is the object

I want to make one thing clear before the rest of my response. Every clause, whether independent or dependent, has a subject and a predicate. If you acknowledge that you have a dependent clause (as you did), then you must also acknowledge an inherent subject inside of it.

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Your assumption is correct. That is why you use the subject pronoun "he" instead of the object pronoun "him".

For example, let's use the most obvious subject pronoun "I" instead.

He intimated that I would go today.

I doubt there is anyone on the forum that would argue that we would use the object pronoun "me".

The reason is exactly what you stated.

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This is your independent clause:

"He intimated that I would go today."

'He' is our subject.

'intimated' is our transitive action verb.

He intimated what?

"that I would go today." is a noun clause functioning as a direct object inside of the independent clause.

"I" is the subject of your dependent (noun) clause, "would go" is your verb group (intransitive), and "today" is acting adverbially to say when you will go.

It is the exact same application for your sentence:

... that he would go today.

"He" is the subject of your dependent, (noun) clause, "would go" is your verb group (intransitive), and "today" is acting adverbially to say when you will go.

Note that I added back "that" from you elliptical noun clause--it makes it easier to visualize the dependent clause.

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I am actually really glad you asked this question because it goes back to a misconception I saw when I first joined this forum:

"who" is a subject pronoun, and "whom" is an object pronoun.

"He intimated who would go."

There are several people on this forum who would argue that it should be "whom" instead of "who", but since "who" is the subject inside of the dependent clause, we used the subject pronoun "who" instead of the object pronoun "whom".