"I was worried about him having AA, KK, or JJ because I opened from the cut-off"
Okay that's fair enough, but as you say, he has AQ, AJ and possibly other marginal hands in his 3bet range. So he could have these in this situation too, and vs your QQ, you want more value from him.
It's a matter of personal preference, but I prefer to strong play JJ and QQ preflop against villains with > 6% 3bet %. The only reason I do is because these hands are so easily beaten post flop by overcards that I want the villain to pay to see the flop. I also want to know how interested he is in the hand too to see if he is playing like he has AA or KK. This is important information to have going into the flop, especially since he will be first to act
If he has AA or KK, he is likely going all in or raising and you fold, not losing any more bets. If he has AK vs your QQ, he likely folds, or calls 4bet, in which case he is paying a lot for blackjack odds
. If an A or K is on the board, you will see him bet on the flop (and you can judge his bet sizing). If he misses then he must risk a substantial bet to convince you he is ahead.
You have the advantage all the way in this spot if you 4bet PF and
if you see a flop.