An Ontario native, Martin played midget hockey with the Toronto Jr. Canadiens before he was originally selected 18th overall in the 2011 OHL Priority Selection by the Mississauga St. Michael's Majors, He made his debut in the 2011–12 season, appearing in 15 games in the last season that the team was known as the Majors. During the campaign Martin won a bronze medal with Team Ontario at the 2012 World U-17 Hockey Challenge, and was also a member of Canada's team at the 2012 IIHF World U18 Championships which won bronze in the Czech Republic. With the renamed, Mississauga Steelheads, during the 2012–13 season Martin was an invited participant at the 2013 CHL Top Prospects Game where he led Team Orr to a rare shutout win, and he helped Canada men's national under-18 ice hockey team win gold at the 2013 IIHF World U18 Championships. Leading up to the 2013 NHL Entry Draft. Martin was rated as a top prospect. He was the fifth goalie selected in the draft, when he was selected 63rd overall by the Colorado Avalanche. In the 2013–14 season, Martin playing in his second campaign as the Steelheads first-choice goaltender appeared in a league-high 64 games and 3562 minutes in posting 24 wins. In his final year of major junior hockey, Martin was signed by the Avalanche to a three-year, entry-level contract on October 5, 2014. While playing in a career-high 50 games during the 2016–17 season, Martin was selected for the 2017 AHL All-Star Classic, the first Rampage goaltender to ever be selected. The same season, Martin made his NHL debut on January 21, 2017, in a 3–2 overtime loss to the San Jose Sharks. Due to the recalls by the Avalanche, Martin was unable to participate in the All-Star Classic. On July 16, 2018, Martin signed a one year contract extension with the Avalanche. As an impending restricted free agent following the conclusion of his entry-level contract, Martin's tenure with the Avalanche ended after he was not tendered a qualifying offer on June 25, 2019. On the opening day of free agency, Martin agreed to a one-year, two-way contract with the Tampa Bay Lightning on July 1, 2019. Martin was one of the eight players called up to the Lightning for their training camp prior to the 2020 Stanley Cup Playoffs.
