Yes you can and it is true, there is a $6K exemption available to NJ veterans.
First use the link that @SweetieJean provided above to submit your proof of eligibility to the NJ Division of Taxation, usually a copy of your DD214.
Then in TurboTax be sure you indicate you are "Former Military" in the "My Info" section.
Next, go to the NJ state interview and after the residency questions, you will arrive at a page titled "Other situations that may apply to you". You will select "Military Veteran" and check the box on the next page as applicable.
You do not have to wait for a response on your submission to file your tax return.
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