Yes, but it is not a direct deduction. New Jersey doesn't list the deduction as a "rent deduction" but rather as a "property tax" deduction. However, you are allowed to claim a credit on your NJ return if you are paying property taxes through your rent. This NJ website gives more information: https://www.state.nj.us/treasury/taxation/njit35.shtml
That website provides that you can either claim a $50 credit (which reduces your tax liability by $50), or you may claim the property tax deduction. In general, NJ stipulates that 18% of your rent is included as property tax. However, there are conditions to this, particularly if you live in a multi-unit building. There is a link on the website with further instructions to calculate the deduction.
Once you figure out which method works better for you, look for the Property Tax deduction/credit, which should be in the "Deductions and Credits" section of the NJ state return.
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