Yes. You can claim a tax credit on your part-year New Jersey return for tax paid to New York.
You will have to figure out how much of the New York tax relates to your eight-months of New Jersey residency. The easiest way is to use the percentage of NY income reported on your NJ return / total NY income x your NY tax liability.
For example, if your reported $80.000 earned while living in NJ / $120,000 total then 8/12 or 67% of your NY tax liability can be used for the other state tax credit.
The tax rate for NY is generally higher than NJ so if you had no other income you generally will not own any tax on the NJ return.
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