Yes, if you TEMPORARILY moved to NY for your internship, but Utah remained your actual home, then you would take a credit on your Utah taxes for the taxes paid to NY.
Utah will tax all income from all sources. NY will tax income sourced from NY (your internship income) What you will do is you will file a nonresident return for NY as a Non-Resident first. Then you will fill out your resident state, claiming a credit for taxes paid to the NY on your Utah return. You may or may not still end up owing money to your resident state depending on whether or not their tax rate is higher or lower than your nonresident state. If the non resident state had a higher tax rate than your resident state, your credit will be limited to the amount of tax you would have paid to your resident state. They will not give you a refund of the taxes you paid to the nonresident state.
Multiple States
File Non Resident State Return
IF you permanently moved to NY, then you would file a Part Year resident return for both NY and UT. You would allocate the income based on where you earned it. If you select Part Year resident in the beginning of the state questions, it will eventually take you to income allocation questions for each state.
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