NJ Resident and Work in NY - No W4 for NJ

You probably don't need to have any New Jersey tax withheld. New York tax rates are higher than New Jersey, so you will end up paying tax only to New York, unless you have other income besides your wages, such as investment income, that is taxable in New Jersey. You will file a New York nonresident tax return and a New Jersey resident tax return. You will pay the full New York tax on your income from the job. Your New Jersey tax return will calculate NJ tax on the income from the job, but you will get a credit on your NJ tax return for the tax you pay to NY, but not more than the NJ tax. The NY tax will be higher than the NJ tax, so the credit will completely cancel out the NJ tax.

Any income you have other than from the job is subject to NJ tax that will not be offset by the credit. If you will owe a significant amount of NJ tax you can make estimated tax payments to NJ. Your NY employer probably cannot withhold NJ tax because they are not a NJ employer. That's why they didn't ask you to fill out an NJ-W4.