Good for you, OP! Climbing gyms are especially good for making friends because you are working on problems with people. My gym has a weekly meet up for people looking for belay partners as well as classes where folks talk. Crossfit might also do the trick, as might a running club. Good luck!
The other thing with climbing gyms, especially bouldering is that you only spend maybe 20% of the time climbing. With 80% time off, that's a lot of opportunities for socializing. You don't get that with the high intensity training like Crossfit where you spend maybe 70% of the time working out and 30% of the time dying.
The gym is not inherently social unless you are actively spotting / alternating uses of a single machine. You either join that group of gym rats (who in my experience spend 80% of their time talking) or you put your headphones on and crack on solo.