You will have to pay tax in NH. They're entitled to their cut of income you earned in their state.
You will first finish your federal return and then do your NH return as a non-resident. After you finish your NH return (which will collect taxes on the income earned from NH sources) then you will do your Massachusetts return.
Massachusetts will give you a credit for taxes paid to other states and deduct what you paid to New Hampshire from your tax bill. Pay close attention to the questions that the TurboTax system is asking as it walks you through al of that. And make sure you do the returns in that order.
@Pepeshto
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