His small bet on the flop may be simply to gauge where you're at in relation to a flop that would be scary for him no matter what he held. He calls your large bet on the turn, which suggests to me that he is not on a flush draw, because even the simplest of players cannot call a bet this size with one card to come. It seems more like he is playing cautiously with a flopped set or overpair. He may also have hit his straight with Q-Q. As far as how to play now, I don't think you need to be worried about being beat by a flush here, because your lagre bet on the turn should have chased away a flush draw. Only specifically the KdQd may have called you here. I would probably push here.