Yes, 4-bet and get it in preflop. Flatting the 3-bet is a huge leak. I think I can count on one hand the number of times in the past 250k hands I played where flatting a 3-bet out of position was correct, and all of them involved playing deep stacked with a speculative hand against a spewy opponent with a very tight 3-bet range. If you don't think your hand is strong enough to 4-bet, then fold. Against a regular, though, 4-betting QQ is essentially never bad.
Postflop, I'd like to hear your reasoning for check/calling the flop before I can tell you what I think you should do on the turn.