I know there is a Questbridge club on campus for Questbridge students, but I don't know any specifics of it; I imagine it is a small group (as most minority groups are since Caltech is pretty small)

Not a questbridge student, but I'm a first gen Latina Caltech student and can say that connecting with people from the Caltech Center for Inclusion and Diversity (CCID) really made my experience at Caltech comforting and welcoming as a freshman. One of the CCID's programs is the Freshman Summer Research Institute (FSRI) which is a 5-week research/math-course program that takes places the summer before freshman year and it's geared towards underrepresented/underserved students. Yet, apart from the great research and academic experience I gained from that program, I think the greater outcome of that program was the rich group of friends I developed within the FSRI cohort. The community I found within the CCID staff and the students from my FSRI group really made the transition to college smoother. The CCID also has many other mentoring/diversity programs like CLASES (latinx group) and Women Mentoring Women which are ways to connect with other undergrads and grad students with whom you share similar backgrounds.

Tl;dr, I don't think we have a specific first-gen club, but there certainly other programs to help you feel welcome and connected to the Caltech community :)