Montour started his hockey journey playing minor hockey in Cambridge, Ontario, prior to joining the Brantford Golden Eagles hockey club in 2010–11. Montour then joined the Caledonia Corvairs for one season in 2012–13. Montour played with the Waterloo Black Hawks of the United States Hockey League and one season with the University of Massachusetts Amherst, where he became the highest NHL draft pick in their school program's history. Montour began his professional career at age 21 with the Anaheim Ducks' affiliate, the Norfolk Admirals of the American Hockey League towards the end of the 2014–15 season. He would then play the entire 2015–16 season with the Ducks' relocated AHL affiliate, the San Diego Gulls. Montour was selected to the 2017 AHL All-Star game for the second consecutive year. Despite not playing the entire season for the Gulls, Montour had amassed 21 points in only 25 games. Montour spent the first half of the 2016–17 ice hockey season with San Diego before he was called up to Anaheim on December 28, 2016. He made his NHL debut on December 29 against the Calgary Flames. Montour scored his first NHL goal in his sixth game against Tampa Bay Lightning on February 6, 2017. He finished the season with 4 assists and 2 goals in 27 games, then provided 7 more assists in the next 17 games of the Stanley Cup playoffs. Montour stuck with Ducks for the entirety of the 2017–18 season and put up a career-high 32 points in 80 games. At the conclusion of the season, with Montour as a restricted free agent, the Ducks signed him to a two-year contract. Montour began the 2018–19 season returning for his third season with the Ducks. Montour led Anaheim defensemen with 25 points through 62 games, however with the Ducks out of a playoff spot approaching the trade deadline, Montour was dealt to the Buffalo Sabres in exchange for Brendan Guhle and a 2019 first-round pick on February 24, 2019.
International play
On April 29, 2019, Montour was selected to make his international debut after he was named to the Team Canada roster for the 2019 IIHF World Championship, held in Slovakia. On May 13, 2019, during a 6-5 victory over host nation Slovakia, Montour suffered a tournament ending lower body injury, he finished scoreless through 3 games for Canada. He later received the Silver medal as Canada progressed through to the playoff rounds before losing the final to Finland to finish with the Silver Medal on May 26, 2019.