I wouldn't even consider betting the turn. Your percieved range is AA, KK, QQ, AK.
Since villain didn't 4bet you it's less likely (although still possible) that he has AA. Means AK is his most likely hand and KK is close behind. Other hands in his range are often small pockets and suited connectors trying to bust a big pair that may not fold to a rag board. So when AK hits the board, it's in your range slightly more often than villain's which means you should bet your QQ and hey, sometimes AJ and AQ will fold right there fearing AK, AA, or KK.
If villain flats you on a board like that with the original better still behind, knowing the type of hands you are likely holding, I'd shut it down after your one stab on the flop. You can pretty much bet the house that villain has either AK, KK, or spade connectors.
Villain has to know you could have AA, KK, or AK, unless he himself is holding one of KK ,AK, or two spades like 98sSomehow is ready to rumble. This screams, AK, KK, or something like 98s
Your turn bet is not profitable long term. Too often if villain is flatting a raise with board texture like that and a decent sized 3bet preflop your QQs are toast when the spade hits the turn. Basically 100% of his range on the turn is now crushing you.
I do, however, still like your raise on the flop because so much of your range smashed that flop. You might even get AJ to fold right there with that bet. If AJ made it to the flop.