To my (American) ear:
- Both sentences are grammatically correct (as TRomano mentioned in the comments).
- The first sentence is redundant.
- The second sentence sounds more natural.
As both Cardinal and TRomano mentioned in the comments, it is normal to say what kind of university degree you received. (This assumes that your listener is familiar with that kind of degree.) For example:
- A bachelor's degree
- A bachelor of arts degree
- A bachelor of arts [with "degree" implied]
- A bachelor's [with "degree" implied]
- A B.A.
"Associate of Arts", "Associate of Science", "Bachelor of Science", "Master of Arts", "Master of Science", "Ph.D.", "M.D.", et cetera have similar variations. ("A doctor's degree" and "a doctor's" are not used. But "a doctorate" is used for most such degrees, with the exception of medical, dental, chiropractic, and legal degrees.)
But if your university gave you a degree that your listener is probably not familiar with, "a university degree" is a good choice of words.
The phrasing in sentence #1 is useful if your degree was in more than one subject. American bachelor's degrees often have "double majors", or "majors" and "minors", or "majors" and "concentrations". For example:
- Fred has a bachelor's degree with a major in information technology, and a minor in music.
- Jane has a bachelor of arts degree, with a double major in Spanish and French.