As a youth, Bibeau played in the 2006 and 2007 Quebec International Pee-Wee Hockey Tournaments with the Victoriaville Tigres minor ice hockey team. Bibeau played in the Quebec Major Junior Hockey League from 2011 to 2014, where in his final year of major junior hockey he led the Val-d'Or Foreurs to capture the 2014 President's Cup and was recognized for his outstanding play when he was awarded the Guy Lafleur Trophy as the QMJHL Playoffs Most Valuable Player. On July 3, 2014, the Toronto Maple Leafs of the NHL signed Bibeau to an entry-level contract. Bibeau was recalled on December 1, 2015, after the Leafs demoted goalie Jonathan Bernier to the Marlies for conditioning. Fellow rookie goaltender Garret Sparks was recalled as well when James Reimer went on injured reserve. Following the demotion of Maple Leafs backup Jhonas Enroth in December 2016, Bibeau was recalled by the team to serve as backup to starting goaltender Frederik Andersen. He made his NHL debut on December 11, in a match up against the Colorado Avalanche. Despite a solid performance by both Bibeau and the team, where they recorded 52 shots to Colorado's 29, the Maple Leafs would lose 3–1. Bibeau's first NHL win came in his second game, where he would make 25 saves in a 3–2 overtime win against the Tampa Bay Lightning on December 29. of the AHL. At the completion of his entry-level contract with the Maple Leafs, Bibeau was not tendered a qualifying offer as a restricted free agent on June 26, 2017. On July 1, 2017, Bibeau as a free agent signed a one-year, two-way contract with the San Jose Sharks. After reporting to his first training camp with the Sharks, Bibeau was assigned to AHL affiliate, the San Jose Barracuda, and posted a 23–14–4 record in his first season with the Barracuda in 2017–18. He had a.919 save percentage and a 2.37 goals-against average, earning a spot on the Pacific Division's All-Star Team. On March 13, 2018, Bibeau signed a two-year contract extension with the Sharks. On October 2, 2018, he was assigned by the Sharks to continue with the San Jose Barracuda. He spent the 2018–19 season in the AHL, where he went 16–13–5 with a.904 save percentage and a 2.89 goals-against average. Prior to the 2019–20 season, Bibeau was traded by the Sharks to the Colorado Avalanche in exchange for Nicolas Meloche on September 27, 2019. He was directly assigned to report to the Colorado Eagles' training camp.
