All the income you earn in Utah is taxable by Utah. The question is whether you will file in Utah as a part-year resident, or as a non-resident. In either case, you must file a Utah tax return.
If you and your husband still maintain your domicile (your permanent home) in Alaska, and your stay in Utah is temporary for purposes of his attending school, then you are non-residents of Utah and would file accordingly.
But if you and your husband have actually moved to Utah, and it is now your permanent home, then you are part-year residents of Utah and would file accordingly. If that's the situation, you became Utah residents on the day you began living in your new permanent home in Utah.
https://incometax.utah.gov/instructions/students
