I'm not much for capping preflop with AKo, but if it is an error, it isn't a large one.

On the flop it's hard to imagine you have the best hand, so take the free card towards your nut flush draw and inside straight draw. If you do bet and get check/raised, call! A flop check/raise is rarely beaten by Ace-high.

Turn is fine, pot is so big and you have a good amount of outs. There are 8 hearts, 3 additional queens, and I'd probably assign 2 more outs to the 6 overcard outs, and 1.5 outs to the 3 queens. That's probably a little generous on the overcard outs though. So you have ~11.5 outs and 16.5:1 odds. Keep in mind you could be drawing dead at this point, but I think a flush or even 89 for a straight and a straight flush draw is a lot more likely than a set or JT.

You made your nut flush, raise! Call a 3-bet, clearly he has at least a boat.