An only child, Conroy was born and raised in Potsdam, New York. He attended high school at Northwood School. His father, Mike, played minor league hockey for the Syracuse Blazers, and at four years old, Conroy served as the team's mascot prior to games.
He followed in the footsteps of his father and two uncles in playing hockey at Clarkson University for the Golden Knights. He enrolled at Clarkson in 1990 and played four seasons for the school. During his junior season of 1992–93, Conroy scored two goals in a 5–3 victory over RPI in the Eastern College Athletic Conference (ECAC) semi-final to help lead his team to the conference championship game. With the goals, Conroy reached 100 career points, making he and Mike the only father-son pair in Clarkson's history to both reach that milestone. Conroy and the Golden Knights then won the ECAC title with a 3–1 victory over Brown University.
The Golden Knights made Conroy a co-captain in his senior year of 1993–94. He led the ECAC in scoring with 66 points and was named to the conference's First All-Star Team, the National Collegiate Athletic Association (NCAA) First All-Star Team and the NCAA Final Four All-Tournament Team. He was a finalist for the Hobey Baker Award as top player in the NCAA, finishing second to Chris MarinucciConroy has a wife, Jessie, and three daughters, Taylor, Sophia and Sydney. Conroy and his family chose to remain in Calgary following his retirement as a player, though they had considered other options including returning to his home town of Potsdam.
He is known for his outgoing and talkative nature, traits that have made him a popular figure with the fans. The Flames honored Conroy with the J. R. "Bud" McCaig Award, given by the organization to individuals that show respect, courtesy and compassion, in 2008–09. He was also named him the winner of the Ralph T. Scurfield Humanitarian Award in 2010 in recognition of his charitable efforts. Additionally, his teammates voted him the organization's nominee for the King Clancy Memorial Trophy which is awarded to the player who best exemplifies leadership and humanitarian contributions to their community. Conroy is the spokesman for the team's charitable organization, the Flames Foundation for Life and routinely participates in team events