Quarshie had considered becoming a journalist before taking up acting. He is a member of the Royal Shakespeare Company, and has appeared in many stage productions and television programmes, including the serial "Behaving Badly" with Judi Dench. He is well known for playing the roles of Sunda Kastagir in "Highlander", Captain Panaka in "", and Ric Griffin on the television series "Holby City". He attended the "Star Wars" fan event ""Star Wars" Celebration" in 1999. He portrayed Lieutenant Obutu in "Wing Commander".