As a resident of New Jersey you need to report all income earned while a resident regardless of where the income is earned. So yes, the K-1 needs to reported to NJ.
I think you are also asking whether to answer "Yes" to the screen "Schedule NJK-1 Received?". If you only received a NY K-1 then the answer should be "No". When you answer "No" the K-1 information will still flow to your NJ tax return from your federal input.
Lastly, you asked whether or not to file a New York non-resident tax return. Per the NY website at https://www.tax.ny.gov/pit/file/nonresidents.htm:
If you are a New York State nonresident you must file Form IT-203, Nonresident and Part-Year Resident Income Tax Return, if you meet any of the following conditions:
- You are a nonresident with New York source income and your New York adjusted gross income Federal amount column (Form IT-203, line 31) exceeds your New York standard deduction.
- You want to claim a refund of any New York State, New York City, or Yonkers income taxes withheld from your pay.
- You want to claim any of the refundable or carryover credits available.
- You had a net operating loss for New York State personal income tax purposes for the tax year, without having a similar net operating loss for federal income tax purposes
