Well I read the OP again and realized for the millionth time I read the hand wrong, as you can see from my earlier post. I thought hero was calling a 3 bet made by the blinds, but instead it's hero that is 3 betting on the button. This changes the entire hand.
I love the flop call as opponent could have a lot of different hands we beat and hands that will give up on the turn. A raise would probably be alright too, but I personally I like to call in this spot with a variety of hands and see what opponent does next. We are in total control with position and the 3bet makes us looks fabulously strong.
The turn, opponent bets the Ace, but that doesn't make much sense. It's unlikely (though would be awesome) that the opponent is betting a queen like this. He may have improved with AK... but even then, he may slowplay the turn. I think that's a pretty common ploy in that spot.
So yes, I do agree a shove is best on the turn. I can't figure out what opponent is repping which usually either means quads or a bluff.