The quartet is set in 2 movements: 1. Allegro (0:00) 2. Rondo: Gratioso (5:45) Published in 1776. Performers: Members of the Danish National Radio Symphony Orchestra.
The painting: the head and the body look like they belong in separate paintings. Portrait painters used to have various set canvases to use when a subject commissioned them, so they just needed the subject to sit a couple of times and they could attach the head to the already prepared body later.
I'm afraid I can't help you in locating a score of that quartet. I believe it as published by Warburton. I'm not aware of a digitized copy of the manuscript accessible online.
MaestroRaro imslp.org/wiki/List_of_works_by_Johann_Christian_Bach You won't find this piece here, but they have lots of others of his pieces (and of all other major composers as well).