It doesn't matter where the organization is located. What matters is where you did the work. Did you do the paid work in New Jersey or in New York?
All your income is subject to tax by the state that you live in, no matter where you earned the income. If you work in another state, the income is subject to tax by the state that you worked in, but you will get a credit on your resident state tax return for part or all of the tax that you pay to the other state.
If all the work was done in New Jersey, you just report all the income on your New Jersey tax return. You don't have to file a New York tax return.
If you got paid for work that you did in New York, you have to file a New York nonresident tax return in addition to your New Jersey resident tax return. Prepare the New York nonresident return first, then the New Jersey resident return, so it can calculate the credit for the tax you paid to New York.
In either case, TurboTax will carry the information about your self-employment income from your federal tax return to the state tax return(s). If you have to file a New York return, it will ask you to identify what income was earned in New York.
