Well you know already this is a terrible flop for your opponent - as in the odds
of him connecting with any of it are very remote (since you have gobbled up 2/3 of it). If he doesn't have a 9 he's out of the hand. So you have to let him catch up - which you gave him that opportunity by checking the flop. If you ranged him pre flop you have him on a small pair to Ax since he couldn't lay his hand down to your raise (which was a good play on your part). So up till then you are doing it right.
You kind of messed up on the turn. A brick hits and your opponent, hoping you have none of this board, throws in a chip check hoping you will go away. After all the flop was paired and the turn was a brick so the odds of you getting any of that is also remote. But you check raise him 6x his bet. Why? He certainly had not improved his hand. Did you really expect him to call, or to even shove?
I am thinking you lost all thought of your opponent because you hit quads and knew you had the nuts and jumped in with both feet - scaring the hell out of your opponent. He took a stab at a small bluff bet and instead of calling and letting him do it on the turn you ran him out of the hand. You lost your patience. Your only real hope of making this hand a big score was for him to hit his A on the river (or whatever he might have), but you did not give him that opportunity.
This is not to say there could have been another brick on the river and you would still miss out on a big pot. It might mean max value is only 02 more cents - but you removed any possibility of getting more by trying to pluck your peach before it was ripe.
So you should have flatted the turn to get an opportunity to get a little or hopefully a lot more.