Yes, according to the Utah Tax Commission, a part-year resident must file a state tax return to report all income received during the time you are a Utah resident.
We recommend that you complete the state interview for Michigan before you begin the interview for Utah. This allows TurboTax to apply any tax credits from your former state to your new state.
If asked to allocate income between the two states, you may base your calculations on the number of months (or days) that you lived in each state. For example, if you lived in Michigan for 344 / 366 days and earned a total of $24,000, you would enter $22,557 as the income for Michigan ($24,000 x 344/366).
Because you work remotely, your income is earned where you provide the service (at home) and not where the company is located.
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