Conroy was born in Vale of Leven, Scotland. He consistently impressed throughout his time with the Celtic youth system and ahead of the 2005–06 season, he signed his first professional contract. In 2007, Manager Gordon Strachan considered calling up Conroy into the first team as cover.
He made his first team debut on 11 December 2007, during Celtic’s 4–0 defeat of Falkirk, playing at left back. After making his debut, Conroy says he hoped he could earn himself a longer term deal with Celtic.
On 6 July 2009, Conroy signed a new deal with Celtic for another year. The arrival of the returning Charlie Mulgrew led Conroy to state that he thought Mulgrew’s arrival would not affect his chance of making a breakthrough at Celtic.