In my opinion, it’s not factually appropriate for your wife to use “first generation graduate”. I have very rarely referred to my self as a “first generation graduate” or the “first in my family to attend college”. When I have, it has been for a specific purpose- to point out the additional difficulties I faced trying to get to college in a family culture * that believed a high school diploma was enough for any man (he could then learn a trade or join the civil service) , and was certainly enough for a female child, (since I would ideally end up being a housewife), and in which my parents couldn’t afford to provide more than minimal support for longer than they had to (a bed , electricity and meals continued through college. My parents stopped providing my clothing and any spending money by the time I was 14) or to point out the almost the reverse- a substantial number of my working class clients ( I did fall for the civil service part 
) were under the impression that since I held a job which required a college education, I must have come from a priviliged background and therefore couldn’t understand their lives. Your wife’s family may not have been financially any better off than mine, or even more emotionally supportive- but having a college graduate grandmother probably means that your wife wasn’t regarded almost as an alien for wanting to attend.
(*Come to think of it, the attitude wasn’t restricted to my family-it was more a neighborhood thing. My parents still live where I grew up, and a few years after I graduated, one of my sisters was talking to a neighbor who said " Doreen graduated, Michele graduated and you’re graduating soon, too".We couldn’t figure out why a neighbor we weren’t particularly close to would know this. And then we realized, out of all the neighborhood kids we grew up with, we were the only three who even enrolled in college)