He is perhaps best known for his role as J. E. B. Stuart in the films Gettysburg and Gods and Generals. Fuqua also appears in the films David Searching opposite Camryn Manheim, Heyday, Something Else, and has an uncredited appearance as Bond Trader Boyce, a stockbroker, in Ed's Next Move.
Fuqua has acted in over 100 plays over the past 30 years, and is the first permanent company member at the Rubicon Theatre Company in Ventura, California. He has acted in 30 plays at Rubicon over the course of 15 seasons (with his 30th appearance in fall 2014). Fuqua received an L.A. Ovation Award (Rubicon's first Ovation Award) for his performance as George in Arthur Miller's All My Sons, directed by Rubicon Artistic Director James O'Neill.
Throughout his tenure at Rubicon, Fuqua has played a variety of characters of different ages, types, dialects, and backgrounds. For Rubicon's 2006–2007 Balancing Acts season, Joseph played Hamlet, under the direction of Rubicon Artistic Associate Jenny Sullivan, for which he received the 2007 Indy Award for Best Actor and the 2007 Santa Barbara Independent Leading Actor Award.
Along with acting at Rubicon, Fuqua is an instructor for their Adult Acting Classes, Fearless Shakespeare Youth Summer Program, served as Associate Director for their production of Jenny Sullivan's original play, J for J, and served as the play's director for the Los Angeles run starring John Ritter. He has directed the Youth Intern production of This is Our Youth, and for Rubicon's Fearless Shakespeare Youth Summer Program, the Summer Youth productions of A Midsummer Night's Dream, Bard On A Wire, Macbeth, As You Like It, The Life and Death of King John, Romeo and Juliet, and King Lear. In June 2015, Joseph directed Hamlet for that program in a three week rehearsal process with Shakespeare doctor and star of Circuit, Jonathan Wade Drahos.
