The quartet was established in 1975 during the studies of one of the members at the Conservatory in Bratislava under the leadership of Prof. Albín Vrteľ. The quartet – still under the name Mucha Quartet (according to the name of its primarius) – attended a master course in Bayreuth (1976, 1978) and one in Weimar (1980). After graduation from the Conservatory, members of the quartet continued studies of chamber music at the Academy of Performing Arts in Bratislava under the leadership of Prof. Tibor Gašparek, and at the Hochschule für Musik und darstellende Kunst in Vienna with Franz Samohyl and Günter Pichler. The quartet adopted the name of the important Slovak composer Alexander Mozyes in 1981. During studies, the members of the quartet received numerous laureate titles of national competitions (3rd prize at the competition Beethoven's Hradec in 1977, 1st prize and the Prize for the best interpretation of the contemporary composition in 1981; Prize of Slovak Music Fund and 1st prize at the Interpretation Competition of Slovak Republic in 1981). In 1980 the quartet successfully represented itself at the international competition Premio Vittorio Gui in Florence; in 1982 it received 2nd prize in the competition of contemporary music and 3rd prize in the main category at the International Competition of String Quartets in Evian, France. In 1987 Moyzes Quartet attended the International Tribune of Young Performers UNESCO in Bratislava. In the same year the Slovak Music Fund awarded it the Prize of Frico Kafenda for continuous promotion of Slovak works and high interpretation level. It received several awards for its recording activities (realization of several dozen CDs and numerous radio recordings).
The ensemble has cooperated with the Dutch quartet Gaudeamus, with Prof. Usi Wiesel from Tel Aviv, Prof. Rudolf Nel from Munich, quartet Melos from Germany, members of the Amadeus Quartet as well as with Janáček Quartet; and from 1989 the quartet has regularly cooperated with Prof. Franz Samohyl in Vienna.
The quartet gained valuable experiences by its artistic collaboration with leading domestic soloists and ensembles (e.g. multi-instrumentalist Jiří Stivín, pianist Marián Lapšanský and Marián Varga, with the ensemble Cappella Istropolitana and many others). It has successfully presented Slovak musical culture in nearly all European countries, in Japan, India, Morocco, USA and Cuba.
Between 1986 – 2005 the Moyzes Quartet was the professional ensemble of the Slovak Philharmonic, in 2006 the chamber ensemble of the town Modra, and during 2008 it became and currently is the chamber ensemble of the town Skalica.
From 2006 the quartet regularly presents itself in its own concert cycles – Moyzes in Moyzeska (2006), Bella in Moyzeska (2007), Suchoň in Pálffy Palace (2008), Mendelssohn in Pálffy Palace (2009), Schumann in Pálffy Palace (2010), Cikker in Pálffy Palace (2011), Zeljenka in Pálffy Palace (2012), Dvořák in Pálffy Palace (2013), J. S. Bach in Pálffy Palace (2014), Moyzes Quartet in Pálffy Palace (2015).
After 40 years of existence in which the ensemble was unchanged in membership, the primarius of the quartet changed, the concert master of the Slovak Philharmonic Jozef Horváth became primarius of the quartet.