[schwza]

i think checking the flop is fine. there's a good chance that the PFR will throw out a bet with AQ/QQ/JJ or some kind of missed hand but will fold if you bet. and if he has AK/KQ/AA, the stacks are too shallow for him to get away from his hand whether you lead out or check-raise.

turn and river are fine. there is no way you should fold anywhere here, and you shouldn't have even really thought about it. the Q is a relatively harmless card. i would be more worried about a T or 5 (that would give an 8 a straight).

btw, limp-calling with KQ is horrible.