No, you don't need to file a NY state tax return just because you received unemployment benefits from NY.
Unemployment benefits are reported to the state where you live, not to the state you got them from (unless you still live there).
If NY withheld NY state income tax from your NY unemployment, and you didn't have any other taxable income, you might want to file a NY return in order to get a refund.
This is from NY States Individual Taxpayer Answer Center
Nonresidents:
Unemployment compensation that is part of your federal adjusted gross income and derived or resulting from employment in New York State is taxable to a nonresident.
If the unemployment compensation received from New York State sources is based on wage or salary income earned partly in and partly out of New York State, figure the amount allocable to New York State in the same manner as the wage and salary income on which it is based.
I would think each state has different rules but appears NY wants you to report - normally you would then take credit on you home state tax return for the taxes paid to NY but in this case NJ does not tax unemployment so no credit would be available.
Not sure what source TurboTaxTodd is relying on - if disagree please let me know, please provide source that says otherwise.