In the early 1980s he appeared in plays such as The Mystery Plays of Wakefield, Three Sisters, On the Wallaby, Pericles, A Month in the Country, As You Like It, Pygmalion, A Hard God, No End of Blame, The Threepenny Opera, Shark Infested Waters, Candide with Nimrod Theatre Company, and a musical adaption of Carmen which he debuted with the Melbourne Theatre Company.
[6] Unable to sight-read music, he walked off the stage at the audition but was called back by Claude-Michel Schönberg and eventually given the part as Javert.
"[6] In 1981, Quast began appearing as a presenter on the Australian children's show Play School, a program he would return to on-and-off again for 17 years.
From 1982 to 1983, he appeared in a recurring role as Dr Rod Hawkins in Australian medical soap opera The Young Doctors for 20 episodes.
Before returning for a second season with the RSC, he spent some time back in Australia, performing in the national tour of The Secret Garden as Dr. Neville Craven – along with Anthony Warlow and Marina Prior.
Quast played the part of Javert on the Les Misérables Complete Symphonic Recording, and in Hey, Mr. Producer, a concert in honour of Sir Cameron Mackintosh.
Though mainly a baritone, Quast has played some roles written for tenors, namely George (see above), Candide, and Archibald Craven in The Secret Garden.
Although he is known for his serious roles, he has also performed comedic parts, such as his 2004 appearance as the pompous Miles Gloriosus in a limited run revival of A Funny Thing Happened on the Way to the Forum at the Royal National Theatre.
Quast more recently played the supporting role of Juan Peron in Andrew Lloyd Webber's 2006 production of Evita at the Adelphi Theatre in London.
In July 2007, Quast performed the role of Judge Turpin in a concert version of Sweeney Todd at London's Royal Festival Hall.
From July 2010, he played Mr. Banks in the Australian premiere production of Mary Poppins at Melbourne's Her Majesty’s Theatre, a part Cameron Mackintosh offered to him in the bathroom of The Ivy in London.
In March 2014, New York audiences were treated to a special limited engagement of Sweeney Todd at Lincoln Center's Avery Fisher Hall.
Quast, in his New York stage debut, performed as Judge Turpin, with Bryn Terfel as Sweeney Todd and Emma Thompson as Mrs. Lovett.
In November 2013 he joined Hugo Weaving and Richard Roxburgh in Sydney Theatre Company’s production of Samuel Beckett's Waiting For Godot.
In 2018 Quast appeared as Arthur Appleyard in the miniseries reimagining of the 1975 Australian Peter Weir classic Picnic at Hanging Rock.
He appeared as Carl alongside Wendy Hughes and Susie Porter in The Caterpillar Wish (2006), and the following year played Ronnie opposite Rebecca Gibney in Clubland.
Quast was in the 2015 political newsroom drama Truth playing real life politician Ben Barnes, alongside an all-star cast including Cate Blanchett and Robert Redford.
He also had a part in the 2016 Mel Gibson-directed war biopic Hacksaw Ridge as Judge, which also starred Andrew Garfield, Sam Worthington, Hugo Weaving and Rachel Griffiths.
He has been noted for his humble nature, stating he doesn't seek after fame and is concerned that success is measured by notoriety instead of the respect of one's peers.