According to Misha Donat’s notes for the Saloman Quarter release of this quartet:
The most famous among Haydn’s six quartets of 1793 is the last, in G minor, whose ‘bouncing’ finale theme has given rise to its nickname of the ‘Rider’.
In German it is the Reiterquarttet, and the nickname “Horseman” is also used. And from the opening bars of the first movement there is a “bouncing”, equestrian quality.
Here is the Kódaly Quartet: playing the First Movement:
This quartet doesn’t make it onto Wikipedia’s page of “program music” i.e. classical music that aspires to explicit representation. Of course, this is a somewhat confused concept in any case. Hayden’s symphonies famously have nicknames, some of which refer to musical features (such as the “Surprise” symphony, which is perhaps not quite the surprise intended if you know its coming) rather than programmatic features.