Call Flop is nice,
i think if villian is thinking he would not
raise the Flop here with a Qx,so his range is
polarized to a set or FDs and Bluffs.
So call him there.
Call turn now again.
If he got a set,he would likely check the turn here,
to give you a free card to improve your draws,so i think
his range is now more Pure Air and Fds,then sets.
Calling the flop here imo is a bit of a mistake, I prefer a fold readless. By calling him we let him dictate to us how big the pot gets and trying to c/c down with TPNK is never going to be profitable long term.
On the turn however, seeing as we got there, I would shove over the top of he's bet, I may just call actually if I think he will barrel again, but I think I like shoving since if he does have a draw we'll either get the pot now or make him put the rest of his chips in with just one card to come. And also we're not going to want to be c/f any river so better to put the money in ourselves now. Shoving just makes things less complicated.
The reason I'm not folding the turn is because it's hard for him to have a hand that he can play so fast. The 4 makes it harder to have a set, and your Q makes QQ unlikely, he may even 3bet QQ pre anyway.
3Bet stats would be useful - is he always 3betting KK/AA for example?, and so would Raise Cbet%, I doubt sample size would be big enough for turn a.f. to mean anything though, although A.F in general may help.