If a sentence contains a complete question, but ends with a statement, should it be punctuated with a question mark?
Could she go to the store, he wondered
I assume you do not want to put the question in quotes, as if the character were saying it, thus:
If you want to describe him thinking, then you could put the thought in italics:
Or you could rephrase the sentence:
It is always possible to rearrange/reformat a sentence to avoid putting a question mark where you don't want it, as well as avoid saying a question with no question mark.