On March 21, 2022, Barron was traded by the Avalanche, along with a 2024 second-round draft pick, to the Montreal Canadiens in exchange for forward Artturi Lehkonen.

[6] He initially attracted good publicity upon arrival to Montreal, specifically by demonstrating proficiency in French as a result of having taken immersion classes during his schooling in Nova Scotia.

[10] Seven days later, the team announced that he would miss the remainder of the season and would be unable to join the Laval Rocket, the franchise's AHL affiliate, for the Calder Cup playoffs.

He admitted that this was initially difficult to process, but after some struggles in his early games with the Rocket, he set about "to play well enough to hopefully be that first call-up, and at some point, get back up there.

[18] Early on into the 2024–25 season, Barron skated in his 100th career NHL game, a 4–3 shootout loss versus the New York Islanders on October 19.

[25] Following the Montreal Canadiens not qualifying for the 2023 Stanley Cup playoffs, Barron accepted an invitation to join the national senior team at the 2023 IIHF World Championship,[26] winning a gold medal.