This is a spot that I cbet almost 0%. Villain gives us credit for an ace a lot, and betting here folds out the hands we beat, keeps in the hands that beat us. Granted there's a flush draw, but he's not going away for one cbet very often if he has that anyway is he?
Checking the flop then betting the turn can extract some value from a queen that may now think they're best, and also will likely lead villain to check river again to us with most of his range, and we can decide whether or not to bet the river, (I wouldn't against anyone who thinks at all, since again it should limit his range to >KK).
As played, I vote for checking the turn. It's WA/WB, which makes me want to check for pot control and keeping his range as wide as possible. If we bet we want either 1) A worse hand to call, 2) A better hand to fold. I don't think a turn bet accomplishes either of these, and basically forces villain to play optimally. A queen will certainly ditch to our 2nd barrell, but that's what we want to call. A flush draw may fold, and forfeit their equity, but they also might call and then we're in a massive reverse-implied odds
spot. A weak ace is the only hand that might fold to our 2nd barrel, but it might actually take 3. And there's also hands that already own us, like two pairs and sets that are never folding. For all those reasons I like a turn check.
When the flush draw gets there and villain leads into us, gotta lay it down.