A university was an institution of higher learning with research and teaching facilities that awarded degrees.

Henry Starling received a Bachelor of Science from a university. (VOY: "Future's End")

Klingon advocate Kolos' mother was a biologist at a university. (ENT: "Judgment")

In 2151, Hoshi Sato taught at Amazon University in Brazil, before she took a commission aboard the starship Enterprise NX-01 as communications officer. (ENT: "Broken Bow") She later briefly thought about returning to the university. (ENT: "Fight or Flight")

Dr. McCoy attended the University of Mississippi in the mid-2240s, at which time he met the joined-Trill Emony Dax, who was visiting Earth to judge a gymnastics competition. (DS9: "Trials and Tribble-ations")

Leah Brahms earned a Bachelor of Science, cum laude, in Subspace Physics from the University of Alpha Centauri, and a Master of Cybernetics in Optical Technologies from the University of Tomobiki in the mid-24th century. (TNG: "Booby Trap")

Mobara worked at the engineering school of a university in the Musilla Province on Bajor. It was here that Mobara was murdered by Silaran Prin in 2373. (DS9: "The Darkness and the Light")

In a scene from the script of DS9: "Emissary", the chancellor of an university on Earth contacted Benjamin Sisko about expecting Sisko's decision, presumably about joining his university.