Are all colleges in Utah Mormon?

No, certainly not, although there are Mormons attending every college in Utah (and in fact, it's quite likely that there are Mormons attending every college/university in the nation, perhaps with the exception of some Catholic or Protestant schools).

The only schools in Utah that are affiliated with the Church of Jesus Christ of Latter-day Saints (the "Mormon" church) in Utah are Brigham Young University in Provo and LDS Business College in Salt Lake City. Other large Universities in Utah, such as Utah State, Weber State, University of Utah, Utah Valley University, Snow College, Salt Lake Community College, Southern Utah University, and Dixie State are all public universities affiliated with the state. Westminster is a large private liberal arts school in Salt Lake City that is also unaffiliated with the LDS Church.