Fleury was selected by the Red Deer Rebels in the second round of the 2011 WHL Bantam Draft. He was named an alternate captain for the Rebels on November 8, 2013. Having been projected as a top 10 pick in the 2014 NHL Entry Draft, Fleury was selected 7th overall by the Carolina Hurricanes, the second defenseman selected in that draft. On August 7, 2014, Fleury was signed to a three-year entry-level contract with the Hurricanes. Fleury was reassigned to the Rebels for the 2014–15 season. On November 5, 2015, Fleury was named an alternate captains for Team WHL at the CHL Canada/Russia Series. Upon a first-round elimination from the WHL playoffs, Fleury was signed to an amateur try-out with the Hurricanes' American Hockey League affiliate, the Charlotte Checkers. On April 12, 2015, in the Checkers final home game, Fleury made his professional debut, scoring a goal, in a 4–3 victory over the Rockford IceHogs. Fleury attended the Hurricanes training camp for the 2015–16 season, then was reassigned to the Rebels for that season. He helped the team advance to the 2016 Memorial Cup, where they reached the Memorial Cup Semi-Final, losing to the Rouyn-Noranda Huskies. Fleury was named to the 2016 Memorial Cup All-Star Team. Fleury started the 2017–18 season with the Hurricanes. On October 26, 2017, in his eighth game in the NHL, he earned his first NHL points, registering two assists in a 6–3 win against the Toronto Maple Leafs. Fleury was re-assigned to the Checkers on January 25, 2018, after recording 6 points, all assists, in 39 NHL games. However, he was recalled back to the Hurricanes three days later. After the Hurricanes failed to make the post-season, Fleury was assigned to the Checkers to help them in the 2018 Calder Cup playoffs. The following 2018–19 season, Fleury again made the Hurricanes' opening night roster, but was re-assigned to the Charlotte Checkers on October 17. He was recalled to the NHL on October 31, only to suffer a concussion on November 23 in a game against the Florida Panthers. Fleury was officially placed on injured reserve by the Hurricanes on November 29. Fleury was activated from injured reserve on December 2, 2018, before a game against the Los Angeles Kings. On July 16, 2019, the Hurricanes re-signed Fleury to a one-year, $850,000 contract extension.
