ITM - JJ with a raise in front.. fairly deep stacks - which way to go?

I don't think this is a good place for a stop-and-go at all. You have a top ten hand; if you're beat right now, QQ/KK/AA is calling your all-in on the flop if it's only unders; the whole point of a stop-and-go is to start with a bad hand in order to get a better hand to fold on a flop, hoping they missed.

Now, as for this hand, I think I'd push here. "Quiet" could just mean a card-dead player, not just tight. And I really don't wanna see an ace/king/queen on the flop and get bet into. I think you're probably getting called by 88+, AJ+, maybe KQs. Depends on buy-in, could be larger range.