Do question marks go after quotation marks?

Sure they could. Pretend you're writing a book and you're quoting someone who is asking a question. Some examples now: "What is Mitch Longley doing these days?", asked Dinie. OR: "Is the Echelon Towers a fine place to live?" asked a prospective tenant. Remember too that the question mark comes before the quotation mark at the end of the question, like in the above examples !