This first movement is indeed conceived with a string quartet in mind. For example, in the forte section at the beginning of measure 1, beat 1, imagine all four players simultaneously producing a forte. Then, the two violins play a duet until the downbeat of measure 3, beat 1, with the viola and cello entering from the upbeat of measure 3, beat 1.

While one might tend to think of measures 7-10 as a duet, taking measure 7 as an example, forcibly distribute the parts among a string quartet: the cello smoothly plays D-sharp and E on the downbeats of beat 1 and beat 2 in the bass clef, the viola continuously sustains B during that time, the first violin plays A on beat 1 and G-sharp on beat 2 in the treble clef, and the second violin plays F-sharp on the upbeat of beat 1 and E on the upbeat of beat 2, alternating with the first violin.

For other sections as well, interpret them personally and always play with an ensemble in mind. Play staccatos like a string instrument's detaché, and consider the right hand in measures 20-21 as a violin's rapid bowing. In that case, the second of two slurred notes should be played weaker, lighter, and shorter than the first.

Try playing everything in this manner, taking inspiration from a string quartet.