The article on the string quartet in Grove (old New, 1980) mentions
early works for two violins, viola and continuo by such composers as
Sammartini and Tartini.  It gives as an example of some relevant
compositions of the time, Alessandro Scarlatti's Sonate a quattro per
due violini, violetta e violoncello senza cembalo, but these were "a
somewhat isolated phenomenon".

Further:

"The removal of the continuo and the adoption of the cello for the bass
line was a gradual and uneven process, though Galuppi followed Scarlatti
in this respect in 1740.

Sammartini's quartets have some importance in the history of the form.
As in his symphonies the first violin dominates the texture but the
melodic line is elegantly shaped in a style in which Classical features
are replacing those of the Baroque.  These three- or four-movement
works,  generally with a minuet finale, were to influence the young
Mozart in his so-called 'Milanese' quartets as well as Boccherini whose
quartets were never to achieve fully the Classical motivic style but
remained in essence a development of Sammartini's manner."

Richard Pennycuick