Conroy was selected in the sixth round, 123rd overall, at the 1990 NHL Entry Draft by the Montreal Canadiens. He turned professional in 1994 upon completing his college career at Clarkson. At his first day of his first training camp with Montreal, he got into a fight with the team's star goaltender, Patrick Roy, after striking the latter with a slap shot in practice. He spent the majority of the 1994–95 season in the American Hockey League (AHL) with the Fredericton Canadiens, scoring 26 goals and 44 points in 55 games. He appeared in six games for Montreal that year. He played his first NHL game on February 15, 1995, against the Hartford Whalers, and scored his first goal one night later against the New York Rangers. Though he was an offensive star at Clarkson and Fredericton, the Canadiens sought to convert him to a defensive specialist in the NHL.