He is known for his long-running role as Ric Griffin on the BBC One medical drama Holby City (2001–20), and for playing Captain Panaka in the Star Wars film Episode I – The Phantom Menace (1999).

He is a member of the Royal Shakespeare Company (RSC), 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 Star Wars: Episode I – The Phantom Menace, and Ric Griffin on the television series Holby City.

He headed the cast of Michele Soavi's The Church (1989) as Father Gus, and played Aaron the Moor in the BBC Television Shakespeare's Titus Andronicus.

[5][6] The episode revealed that Quarshie is part of his country's old mixed-race elite as one of his ancestors, Pieter Martinus Johannes Kamerling, was a Dutch official on the Gold Coast.