Fleury was drafted in the fourth round, 87th overall by the Kootenay Ice in the 2013 WHL Bantam Draft. Fleury began playing with the Ice in 2014, and was named team captain during his third and final full season with the team. On November 11, 2017, Fleury was traded to the Regina Pats to help bolster their roster in hopes of going on a deep playoff run.[1]
Fleury was selected 87th overall by the Montreal Canadiens in the 2017 NHL Entry Draft in Chicago. On October 4, 2018, Fleury signed a three-year entry-level contract with the Canadiens.[2] He was assigned to Montreal's AHL affiliate, the Laval Rocket, where he played his first season at the professional level.
On October 1, 2019, it was announced that Fleury would appear in the Canadiens' opening night roster.[3] On November 16, 2019, Fleury earned his first NHL point, scoring a goal during a 3–4 overtime loss against the New Jersey Devils. He split the 2019–20 season between the Canadiens and their minor league affiliate Laval Rocket of the American Hockey League.[4]