Question mark usage/position when sentence ends with a declarative quotation?

Who said "I am hungry?"

The quotation is not a question, but do I put "my" question mark there anyway?

This is related to, but not the same as Position of question mark when sentence doesn't end with question

If you put the question mark inside the speech marks, the speech becomes a question. Your example is thus incorrect as 'I am hungry?' makes little sense as a question, unless the person has brain damage and is unaware of his hunger!

A more valid example might be:

He asked, "Which way to the museum?"

If you shift the ? outside the speech marks:

Who said, "I am hungry"?

Then the whole sentence is now the question: the speaker is asking who stated that they were hungry.

You only use one punctuation mark when using speech/quotation marks.