You almost have to bet the flop, but at least the check threw him off the scent. The 3-bet on turn could be an A or maybe he has a Q, or maybe nothing at all--again because of the flop check, he's going to put pressure on you to fold.
If you're really afraid of QQ don't be. Raise the roof here, to at least give him a CHANCE to fold.
Most of the time you're chopping--assuming he actually has a hand and calls you down. Once in a blue moon you're beat but those are the breaks. If it was me I'd hope he had AQ just to rub it in on his horrible slow play.
Peace.