Hi,
Check/fold: Good if players are expected to play super honestly, and you don't really want to turn QQ into a bluff, and when someone bets you think he is rarely bluffing.
Check/call: Good if the players are expected to play more honestly, and if you are viewed as capable of having a balanced checking range and are also capable of making huge hero-calls so they are less likely to run crazy bluffs here.
Bet: Good if we are vs 2 regs who will put us in tough spots. In general I would often cbet this Ace high board with air, and if c/c or c/f aren't desirable then I would just opt for this option instead.
What's the best option here? I honestly wouldn't mind checking and deciding. If BB bets and we know he is a little aggro and CO folds, maybe we can call here. If CO bets we are in a tough spot since we are not closing the action. But given how the hand played out I'm not giving him too much credit for having an Ace himself. We block AQ with our 2 Qs, he would 4bet AK preflop, and he probably folds AJo here. I would think that he is repping really narrow here.
That would be my personal preference (check/decide, probably to call) though we can c/f if BB bets and CO calls. But this is part of my usual game plan (to have a wide check/call range in 3b pots).