As a native U.S. speaker, I would say these things, in this way, using this phrasing:
a) I am a college graduate. (That is, I went to a university for four years, and didn't flunk out).
b) I have a bachelor's degree from [X university].
c) I went on to grad school at [Y university].
d) My undergraduate degree from X was in Z. OR,
I majored in Z at X].
e) My graduate degree is in Q. OR,
I got [a master's degree/a doctor's degree/a Ph.D.] in Q.
I wouldn't actually use the term "postgraduate." I see "postgraduate" mostly on questionnaires where I am being asked to indicate how much and what kind of education I've had.
I can imagine this dialog:
"Have you had any postgraduate education?"
"Yes, I got a graduate degree from the University of Winnemac, it was a master's in Q..."