We talked about the Arabic she is studying at university

Your first sentence should be "She studies Arabic" no "an" because there is only one Arabic language and then "language" is understood. Then in America you would say: "We talked about her study of Arabic in college." We use college to cover both colleges and universities in this context. This phrasing would be more specific to the course vs. the language. As in, why she is studying Arabic, who is teaching, how is it going, etc.