44 full at flop

Slowplay flop is totally ok to balance, specially on paired boards.
Turn raise is ok too, you should try to get as much value as you can. You must call the 3bet shove, both of you seem to have very decent hands (A3, QQ, maybe some overpairs as well).

Obviously I shove but my opponent said I played badly that hand.

He had QQ but I think he would not fold if I raised at flop. And I think I miss some value if i raised the flop.

I would play the same way. Bluff raise the flop with so fee combos in our range in BTN vs CO is not the best option imo. Just flat and keep both ranges wider.
I think your line is ok. The Q is a brick but he has shown enough strength to justifify a turn raise and still be called.
Besides, his turn bet is so small that you have to defend your whole range basically.

You play perfectly well, flop and turn. If you have some sick read you could fold on turn, but its very hard on NL2. Thats just standard play.

You lost the hand obviously. If he has QQ or 33, or hit something OTR thats just a cooler, you can do nothing. I think it is well played.

Decided to solve this hand (I don't know if I'm allowed to write the name of the software, but it doesn't matter much).
Main conclusions:

- Villain should be double barreling more than 75% of the time. About 55% of the time using a 1/4 bet, almost 30% overbetting and some 1/2 bets (16%). It was what villain did here in fact.
- When villain bets 1/4, we should mostly flat call. The exceptions are flush draws, which we should be raising with about 50% of the time. TT and 99 without a heart (which I decided to include in your range to flat preflop) should also be raised often, flat call with a heart mostly.
44 and 33 should be raising about 60% of the time.

- When you raise villain must be flat calling about 70% of the time. He should shove AA/KK and AQ, A3 here.
When he shoves, you should be calling with 44, 33, AQ, even QJ, QT, Q9 (and the nutflushdraw too). Obviously, this assumes villain can be bluffing with 65s, 54s, 75s, A2s, A5s.
I believe it is going to be mostly value the 3bet shove here, so probably remove some Qx from your calling range to a shove is a good idea.

- Villain should be flat calling QQ 100% of the times to your raise.

Thank you.

Wich program are you running ?