I was wondering is the form of the string quartet's first movement usually sonata-allegro form?
Yes, and the same applies to all movements. See
http://en.wikipedia.org/wiki/Sonata_form#Sonata_form_and_other_musical_forms
The form of the first movement of a string quartet is usually sonata-allegro form. In fact, this applies to all movements of a string quartet as well.
To find this information, you can refer to reliable sources such as music textbooks, scholarly articles, or reputable websites. One such source is Wikipedia, which has an article on sonata form that discusses its use in various musical compositions, including string quartets. The article explains the structure and characteristics of sonata-allegro form and provides examples of different works that use this form. By consulting such sources, you can gain a deeper understanding of the form and its application in different musical genres, including the string quartet.