Perflop
Not 3-betting QQ is pretty conservative, but it is a battle between the first two players on the table, so I can get on board with an occational flat, especially if you have aggressive squeezers behind. If nothing else it gives you a hand, that can easily defend against a squeeze. I am also ok flatting against a nitty UTG, like a 15/11 or something, if you are using a HUD.
Flop
When he check to you, I would put out a small bet like 30% pot. I think, him checking indicate some kind of showdown value, and while checking back can look weak, I am not sure, how many turn bluffs it will really induce. If the guy wanted to bluff, would he not just have C-bet on this incredibly dry board?
That being said I would rather check back QQ than a hand like 99, since it needs less protection. There is also some risk he is checking a "trap" like AK, and getting check-raised would be absolutely awfull. So I guess, checking back is ok, especially if you think, something like that might be going on.
Turn
Obviously the one card, you did not want to see, and now I am definitely not happy to face a bet. If you are behind, you only have two outs, and you can certainly have some AX, that check back the flop, so you have better
hands to defend with than this one. You also have bad removal not blocking his value but blocking some of his most likely bluffs like QJ or QT, which just picked up a gutshot.
River
Another ace makes it less likely, he has one, but on the other side any AX is essentially the nuts now, so AX is certainly always betting again, and there are still many combos of it in his range. So all in all I would also fold now, but I think, it would have been slightly better to fold on the turn.