I was taught that if the quote has more than one sentence, then you do it like this, with a period in and a period out:
Statement:
I once heard a quote that said "Stop. You can't go. If you go, you can't stop. If you stop, you can't go.".
Question:
Is there a quote that says "Stop. You can't go. If you go, you can't stop. If you stop, you can't go."?
That isn't a real quote, by the way.
If it only has a phrase, or if it only has one sentence, then you do it like this:
Statement:
John said "Get out of here".
Question:
Did John say "Get out of here"?