Conroy has set his sights on a new contract after making his first-team debut in the 4-0 victory over Falkirk.
The 20-year-old replaced Darren O'Dea at left-back as Gordon Strachan opted for a player more suited to the role than the Irish central defender.
Conroy, after some nervous early touches, grew in confidence and linked well with hat-trick hero Aiden McGeady down the left.
The Scotland Under-20 international now hopes to keep his place for Sunday's trip to Inverness, and prove he is worth retaining beyond the end of the season.
"Hopefully, if I keep working hard, I will get a new contract," Conroy said. "No-one has said anything to me yet, but it is still early so I will just concentrate on playing my football.
"I was delighted to get a start, it was a real surprise.
"I thought when Lee Naylor got injured against Shakhtar the other week I might get my chance as I play that position in the reserves. Now it has come and I want to take it."