Yes, the university can support you for the green card process. Please check with the university about it willingness to file for you.
The GC process is initiated through PERM which involves a "pattern of recruitment" for U.S. workers. This procedurally complex stage of the green card process directly involves the employer. Hopefully your university agrees to support you.
This is a complex process. Generally, the H-1B is a great temporary visa to have while you are processing toward a green card. Speak with a lawyer to see if you qualify.
Kenneth Craig Dobson has practiced immigration law for over eight years. Nothing in this response should be considered legal advice. To be clear, no attorney-client relationship is established unless and until a written engagement letter is signed by both parties (attorney and client).