Billquist graduated from Göteborg city theater in 1957, and was hired at the theater after his graduation. In 1964, he was employed at the royal theater in Stockholm, Dramaten, until his death in 1993. His stage credits include Woyzeck, Richard III, and Pinocchio.
From 1961 to 1963, he could be seen in TV theater productions made by SVT. His first movie credit was a short film called Lucia 1959? His most notable film work, however, was with Ingmar Bergman, as the young man in All These Women (1964) and years later as police superintendent Jespersson in Fanny and Alexander.
His voice credits include The Smurfs, The Smurfs and the Magic Flute, Pelle Svanslös, Les Daltons en cavale (as mountie Pendergast), Tom & Jerry: the Movie (as Dr. Applecheeks), and others.