No - If you are a resident of NY but receive NJ unemployment income, this unemployment is not taxable in NJ. However, Unemployment compensation is taxed by New York State. Report the same amount on your New York State tax return as on your federal tax return. If you did not file a federal return, report the same income you would have reported for federal income tax purposes
According to the NJ Department of Revenue -
Unemployment compensation and temporary disability (including family leave insurance benefits) received from the State of New Jersey or as third-party sick pay are not taxable in New Jersey.