Don't have a really good answer as it is kind of a gross spot, but just gonna speak to the math. So you say that you have odds to call on the turn, which you do, i count 10 outs for Quads & FH, so like ~20% equity and its slightly less than that on the call. However, you say that you don't want to re raise on the turn because you run into flushes a lot, which COULD be true, however if you are gonna play by math then you are kind of stuck, I think. The only reason you would only call the turn is so you can get away from the hand on the river, but you don't, and that is because after the river shove you only need to be right on the call 20% of the time with the odds you are getting.

So as far as I can see, you have 2 options. First, play the math, in which case, I think you just need to jam the turn because you are gonna be calling the river anyway, if I am calculating it correctly. Second is play kind of exploitative, in which case, you fold at some point and if you are gonna do that, cuz you think this guy has a flush most of the time, then why not just do it on the turn, when he raises, and save yourself 26bb. Maybe someone can make an argument that there aren't enough hands that you beat for every hand that you lose to on the river shove, that you can fold it there and its not exploitable and it is mathematically correct as well.

I personally think I would have just jammed the turned, and hoped I had the best hand or hit one of my outs. Since it is 2NL though you can probably make folds on either the turn or river and not have to worry about being exploited.

I actually had a hand almost identical to this today. I had QQ and flopped the set with monotone board as well. I bet, he called flop. I bet turn, he raised, I jammed, he called. This was in a 3bet pot and he turned over 56 for a pair of 5's with a straight draw on the turn and I held river. I think these guys have the capability of showing up with all sorts of nonsense. This was on 2NL as well.

Pre flop at 2NL I much prefer a 3 bet in position with QQ

I like the raise size on the flop andI'm going with it too.

On the turn I would carry on with the same plan and get it in. Villain is probably stacking off all his flushes once you raise on the flop so it looks like he's trying it on thinking you're on a bare club draw.

As played on the river you have to call with those odds