It depends, but usually military pay is taxable in New Jersey. The following information, taken from the New Jersey website: (Click on following link for full article: Military Personnel)
Resident and Nonresident Defined
Residents. If your home of record (and domicile) was New Jersey when you entered the service, you remain a resident of New Jersey for income tax purposes, unless you qualify for nonresident status (see below). Your domicile does not change when you are temporarily assigned to duty in another state or country.
If you are a member of the Armed Forces whose home of record (and domicile) is New Jersey, and you are living aboard ship, in barracks, billets, or bachelor officer quarters, you remain a New Jersey resident for income tax purposes.
You are not considered to be maintaining a permanent home outside New Jersey if you are living in such accommodations.
However, if you pay for and maintain, either by out-of-pocket payments or forfeiture of quarters allowance, an apartment or a home (either owned or rented) outside New Jersey, such facilities will constitute a permanent home outside New Jersey. In this case you will no longer be considered a New Jersey resident for tax purposes.
If you are a member of the armed forces and your domicile is New Jersey, you are considered a nonresident for New Jersey income tax purposes if you meet all three of the following conditions for the entire year:
Based on the above definition, NJ will not tax your income if you maintained a permanent home outside of New Jersey, but you must forfeit your quarters allowance to do so. Otherwise, your military pay is taxable in NJ.