Quilty was born in Ottawa, Ontario. He played junior hockey with Glebe High School and the Ottawa St. Pats of the Ottawa City Junior League. He became a professional with the Montreal Canadiens in 1940–41, winning the Calder Trophy. After two seasons, he joined the Canadian Air Force where he kept active in hockey playing on RCAF teams in Toronto and Vancouver. He returned to professional hockey with the Canadiens in 1946–47. He played part of the 1947–48 with the Canadiens before being traded to the Boston Bruins. He suffered a compound fracture of his leg that season and did not return to the NHL again. He played senior hockey for several seasons afterward for North Sydney Victorias , the Ottawa Senators and Renfrew in 1951–52.