The word "university" is reserved for institutions of higher learning in many (most?) states under their state education laws. The use of the word "university" can also be limited by state consumer protection laws. Trademark law, however, does not prohibit its use in a name.
To give a widely-publicized example, Donald Trump in 2005 created his
Donald Trump University, an online "education" center which offers online real estate and management classes. It is not accredited, there are no college credits available for transfer, and no degree is proffered. Shortly after its creation, the New York Department of Education (along with Maryland's Department of Education) informed Trump that his use of the word "university" was both "misleading and illegal" under state law. Trump was forced to change the name of his online class offerings. His "university" is now called
Trump Entrepreneur Initiative.
As an aside, student complaints over Trump's online classes have led to investigations into its operations and has spawned at least one civil suit. As another aside, "Entrepreneur Magazine" is trademarked and
Entrepreneur Magazine has sued in the past (unsuccessfully) over the use of the word "entrepreneur" by others. I don't know if they have attempted to sue Trump, however.
Generic names on their own cannot become registered trademarks. Words like university, college, bank, clinic, church, and law office cannot on their own be registered by any one company. The trademark owner if allowed to have such a mark could potentially prevent others from using these words in their names. But, even though university cannot become a registered trademark and would therefore be free to use without infringing on the rights of another, there are
other laws that may come into play when generic words such as these are used for a business. Prohibitions and restrictions may apply.
When naming your business, you will want to be careful not to infringe on anyone else's trademarked business name, so this will take some investigation on your part. You also need to be aware of laws, both state and federal, that may affect the naming of your business. And you must avoid misleading or deceiving consumers into thinking your business is something it's not.
I strongly recommend that you look for a name for your business that does not include the word university. Consulting with an attorney in your area is always wise and advised when starting any new business. A new business is challenging enough without adding in any unnecessary legal complications.
Good luck.