Graham was drafted 155th overall by the New York Islanders in the 2012 NHL Entry Draft and signed an entry-level contract with the team on May 31, 2014. He spent three seasons with the Islanders' American Hockey League affiliate the Bridgeport Sound Tigers, as well as spells in the ECHL for the Florida Everblades and Missouri Mavericks, before signing a one-year contract with the Colorado Avalanche on July 26, 2017.