Moved from NYC to NJ; Employed in NYC

Yes, if it is a permanent move you will file a Part-Year resident return for New York and New Jersey.  In general, you pay income to the state you worked in as a nonresident, and report all income to your resident state so:

Your New York Part Year return will have all your wages: for January to September, it will be resident income and from October to December, it will be nonresident New York source income.

For your Part Year New Jersey income, you will report the October to December income.  You will claim a credit for the taxes paid to another jurisdiction for this income that was taxed in New York as nonresident New York-source income so you will not be double taxed on the same income at the state level.

If you have questions as you are entering the information, please add to this post.