For 2019, yes. Your income is taxable in both states. The income you earned in New York is taxable as a nonresident, (because it was earned there) and the same income is taxable in New Jersey as a resident because you lived there when you earned the income. To reduce the effects of double-taxation, New Jersey will provide you a credit for the tax you pay to New York.
In TurboTax, prepare the New York return first. You will then select the New Jersey part-year resident return. Watch for the screen About this Income in the New Jersey interview, and make sure that the box for New York is checked to make the income not taxable in New Jersey. Later, when the program asks about double-taxed income, you should see the income reported to New York and the amount of New York tax you are paying on that income. If you see amounts there, the credit will be calculated on the tax reported.
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