Retirement tax questions

Yes, if you followed the directions given in this Turbo Tax post, the amount shown in Box 5A is not taxable but it shows up as a taxable distribution in the NJ-1040. This is because New Jersey does not allow non-deductible IRA contributions according to this post from Nj.us. As a result, you will need to enter  information in the NJ tax return that would deem this distribution non-taxable.  

Remember with  a Back Door Roth, it was originally a contribution that became a distribution in the 1099R.  From the distribution, it became a conversion, all done through the back door.

As you complete your New Jersey return, there will be a screen addressing your Roth Conversion, here you will record the amount of $6000 as a contribution previously taxed by New Jersey. Here is what the screen looks like in the New Jersey state tax section.