I'll typically want to slow down with QQ on an A-high board so I'd often just start with a check. Most of your value with this hand is going to come from either bluffcatching or getting a small amount of value out of worse hands (1 or 2 streets, max). As played whichever river line you take is going to be very meh. Calling for those odds seems good but it requires villain to have some random floats on the flop or be betting something like TT or 8x on the river which I think should be rare.

Second pair type hands make a better check than a bet in single raised and 3 bet pots. It's difficult to get called by worse and checking helps us by including bluffs in his range on the turn. We're also probably way ahead way behind sogiving a free card doesn't hurt much here.

The turn is grim because we can't really bet again for value but don't want to let villain freeroll a club on us. Checking is probably best but I would fold the river.