No, from your description, you would not be required to report New York income, nor file a New York return.
HERE is a link for what New York considers New York Sourced income.
If Box 5 and 7 on Form 1099-NEC are blank, I assume Box 6 simply reports the company state ID and is included on all the forms that company uses.
If TurboTax gives you an error for this (it shouldn't) you can leave box 6 blank as well since the federal return does not need the state ID number.
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