There are two issues here, both probably individually simple, for different reasons. It appears that ‘the food was all gone’ is something that Bloggs says, and that your full example is questioning Bloggs’s accuracy.
In fact the question mark itself turns out to be something of a red herring. There is nothing about this example involving a question that helps to resolve it, and in the end there is also a case for disposing of the question entirely.
First, there is no reason to place a question mark within your quotation marks. Bloggs simply is not asking a question, and it would be academically fraudulent (misrepresenting the original) to insert a question mark as if that had been Bloggs’s intent.
Second, we have a matter of style as prescribed by the venue where you are writing, concerned with referencing at the ends of sentences, regardless of whether nor not any given sentence is a question. Your publisher, university department or whatever will have a preferred style guide for this kind of thing.
My own tweaked form of the MHRA style guide (which allows for considerable latitude if used consistently) would in fact use footnotes (pp58ff), with the footnote number appearing after whatever mark closes the sentence (full stop, question mark, quotation marks, anything). On occasion that can suggest rewriting a sentence so that the footnote mark does not end up separated from its referent by a pileup of punctuation.
I mention that partly as a matter of contrast from your format, which is a form of Harvard referencing. On that basis your suggested inline reference would be fine, and would be inserted within the sentence; i.e. before the question mark. (Liverpool University provides a usefully succinct guide.) That leads to the correct but awkward form:
Was it true that ‘the food was all gone’ (Bloggs, 2013, p. 287)?
That looks horrible. If this were my paper, I would therefore rewrite (including preferred rules about punctuating quotations of complete sentences) to something like:
We must therefore question the accuracy or sufficiency of Bloggs’s
ostensibly simple statement that, ‘The food was all gone’ (2013, p. 287).