Yes. That's one line you could take with your hand. The idea is to try to win it right there.
As you played it, appearing to be drawing by just calling, you could have bet 1/2 pot on turn representing a straight. Here again you had a chance to win it right there, and still had 9 cards to your flush draw, and 3 more Q's to your straight draw, plus 2 more A's ,plus 2 more J's. for 16 outs . This is a great semi bluff opportunity.
I'm not big on pushing though. You'll get that advice often. IMO don't take it. If you do, the entire table will soon be slow playing their hands to trap you.