No explanation required: plenty of colleges / universities are mentioned in Star Trek.
Here is a listing of all of the colleges, universities, and degree-offering institutes mentioned in Star Trek television series and films.
Real universities and colleges (i.e. ones that exist today on Earth):
- California Institute of Technology
- University of Cambridge
- Indiana State University
- Kent State University
- Princeton University
- Stanford University
- Trinity College Dublin
- University of California at Berkeley
- University of Manitoba
- University of Mississippi
- University of Nairobi
- University of Texas at Austin
Universities and institutes that only exist in the Star Trek universe:
- Denebian Academy of Science (Deneb IV)
- Aldebaran Music Academy
- Andorian Academy
- Banean Engineering Institute
- Central University of the Cardassian Union
- Daystrom Institute
- Regulus III Science Academy
- ShiKahr Academy
- Starfleet Academy
- Starfleet Medical Academy
- University of Alpha Centauri
- University of Bajor
- University of Betazed
- University of Copernicus
- University of Culat
- University of Hurkos
- University of Orion
For the exact appearances of each of these universities and institutes in the Star Trek universe, see Memory Alpha here.
