No, if none of your income was sourced from New York, there would be no filing requirement for you.
See Who must file? on page 7 of 2020 New York Instructions for Form IT-203 Nonresident and Part-Year Resident Income Tax Return
Your wife's school tax credit will be limited. See page 29 of 2020 New York Instructions for Form IT-201 Full-Year Resident Income Tax Return
@gs77
