Live 2-20 Spread Limit: Flop a Boat turn Quads

[SuitedSixes]

This is a live 2-20 spread limit hand. Blinds 1/2

I am in UTG +2

UTG limps
I raise to 8 with QQ.
Tricky player MP1 who likes to isolate against me makes it 18.
Folds to tricky/aggressive player on button who makes it 38.
I call.
MP1 raises to 58.
Button calls.
I call.

Pot is $158.

Flop comes Q44.
I lead for $20

MP1 folds

Button calls.

Turn is a Q.

I lead for $20

Button folds.

**I have been making a conscious effot to play as straightforwardly as possible. At the time, I felt that at least one of these guys has AA or KK, so I was looking to be raised on the flop. Should I have check-raised the flop? How about the turn? What's the best way to get value here?