I think he has a Q most of the time here. He PFR and bet out on the flop so the only semi-likely flush draws are AKs, AJs. Having said that, if you call and he has overcards he is pretty unlikely to bet into you on the turn with a Q there. If you minraise him and he has AQ, KK, AA he probably raises again to find out where he is (or at least calls) so the minraise probably works out +EV. If he has overcards then call, check call/bet might work but Id go with the raise. I think you can rep a Q by raising the flop, whereas the flop-call, turn-raise line betrays alot of strength (especially if the flush card comes on the turn and he decides you were on the draw).

I think I play it like you did but im not scared of the flush draw. I just want more value from a real hand.