Just to confirm before you read too much into this I am pretty much 'begginer' so this is just how I would have played this, any feedback on my comment is welcome.
Its a tough hand, if in position OTF I may raise as it means if he is only one pair hand you may get to see river for free, but if you check the turn, expect a big bet on the river from your opponent. You could represent AK of spades in this scenario unless obviously he has the A or K of spades in a monster pocket pair. But this board is scary as the flush draw is on, his Aces or Kings may not look so good if you 3Bet the flop. He also 4bet preflop, he wouldnt be doing this with pocket 6s, 4s or 9s? So i would rule he has a set here and continue on the premise that he has QQs+ and AK/AKs. Which means he more than likley has you crushed so would a 3Bet actually work here?
With JJ I tend to play it as a set mining hand, if OR and someone 3bets I call, if i completely miss the flop i fold but if lower cards on flop i may continue or if there is some chance of a drawing hand coming on the turn with only 1 over card i may also continue but it depends on opponent.
In this situation QQ you are good to 3Bet with obviously, but when 4bet I would probably play it as you did (call to flop), but with such a low flop I would at least call here and re-evaluate my hand on the turn and play it then similar to JJ, if turn comes scary card or you dont hit a Q and he bets again i think its time to lay it down.
If you call and raise you need to have a plan on the flop for what you are going to do when the turn card comes, if its lower again and he bets 2/3 pot again he is showing strength and it may be time to fold.
But with the raise and fold if seeing any further aggression from your opponent, when you actually make a hand next time and play it similar to this he may get it allin the middle and you may take down a massive pot.
