Yes. You will need to file a non-resident return for the other state to report the income earned during your internship. The non-resident state will only tax you on the internship income.
You will also file a return with your resident state to report all income (including the internship). Your resident state will allow a credit or adjustment for any tax paid to the non-resident state. (You will not be taxed on the same income by both states.)
In TurboTax, after completing your federal return, move on to the non-resident state before doing the resident return. It is important to do the non-resident return first so the credit or adjustment can be calculated and applied to your resident state return.
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