The group was founded in 1931, uniting four soloists from the Bolshoi Theatre Orchestra. They gave their first concert on December 11th, and soon took the name The State Quartet (GABT). They were sometimes informally referred to as the 'Gabtovtzi'.
The original lineup was:
Isaak Zhuk - First Violin
Boris Weltman - Second Violin
Moris Gurvich - Viola
Isaak Buravsky - Cello
They gained fame throughout the 30's with extensive radio play. They won joint-first prize at the Composer's Union Quartet competition in 1936, sharing with the Komitas Quartet.
Shostakovich wrote their praises in 1938, saying:
"The artists of the Bolshoi Quartet are notable for their individual qualities and as a whole present an excellent ensemble that shows high musical culture. Their performance of Debussy's quartet was excellent, and equally outstanding was their Haydn. Since Haydn's and Mozart's pieces are commonly the weakest spot in the repertoire of string quartets, this performance is noteworthy."
Shostakovich performed with the group several times, playing his own Quintet, Op. 57.
The Bolshoi Theatre Quartet and Shostakovich
On 18th March 1945, the Bolshoi Quartet premiered Weinberg's Piano Quintet, Op. 18, with Emil Gilels. Weinberg's Fourth Quartet is also dedicated to them - they premiered that work on 19th January, 1946.
The group went on to make many notable recordings, focusing on 19th and 20th century Russian repertoire. They disbanded in 1968.
I shall be updating this post with more info as and when I come across it.
N.B. - picture and Shostakovich quote can be found on Valentin Zhuk's page: Link