I received a 601B rev 05/20 from the state of New Jersey that I owe $871.70.the state of New Jersey is stating I/O $871.70 for my 2020 state income tax return I thought that I had a guarantee

You should have received a letter explaining the reason for the increase in tax. What was the reason?

Was that the original tax due or did the state change your return?  If that was the original tax due, how did you pay it?  Turbo Tax never pays any tax due.  They only pay interest and penalties if there was a calculation error on your return.   Did you think the state tax due was suppose to come out of your federal refund?  That is not possible.

Check the printout or PDF of your return; look for the state cover sheet with the Turbo Tax logo. If you owed tax, it will show the payment information and how/when you decided to pay. Read the state payment instructions carefully since most states cannot be paid from within the TT system and requires additional steps.
https://ttlc.intuit.com/community/refunds/help/what-are-my-options-for-getting-my-state-refund-or-for-paying-any-state-taxes-i-owe/00/25954

@tom-leonhardt 

This is a public web site that can be seen by anyone—including scammers and would-be identity thieves who would love to contact you and pretend to be from TurboTax.  Please remove the personal information you posted here ASAP by returning to your post and clicking the three little blue dots on the upper right to edit your post.