It depends on your usage.
If you are quoting someone that is asking a question then yes the question mark should go inside the quotation marks.
For example:
- “Are you going to the party later?”
- “Did you say you wanted steak for dinner?”
- “Do you like listening to music?”
But, if the item in the quotation marks is simply citing something someone else has previously said, the question mark doesn’t need to go inside them (unless of course it’s a question).
- Are you telling me that he really said “go away”?
- Did you definitely hear him right when he said “take the second left”?
- Did she really say “I’m leaving town”?
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