Can TurboTax Live pay NJ state taxes online? I'm asking this question because form NJ-1040 does not have any place to specify bank account info.

You can make a NJ tax payment online at the NJ website below -

https://www1.state.nj.us/TYTR_RevTaxPortal/jsp/IndTaxLoginJsp.jsp

Yes, I know that I can make NJ tax payments online or by sending a check which obvioiusly has nothing to do with e-filing returns by TurboTax.  I had provided my bank account details to TurboTax and it was mentioned in the "cover page" of our e-filed NJ tax returns (PDF file with form NJ-1040, etc.). Does that imply TurboTax will be passing on my bank account details to NJ Taxation?  I don't want to pay taxes 2 times -- once through TurboTax e-file and another time directly via online or via check myself.  On the other hand, if TurboTax didn't pass on my bank account details, we will considered delinquent even though we e-filed our tax returns on time.  Hence the question.

Yes, if you efiled the NJ return and provided the bank account details for payment (which it appears you did since its on the NJ cover page), then this information gets forwarded to NJ for payment on the date you selected.  There is no place on the NJ form because NJ requires efile for direct deposit/payment.  So, if you were printing your return, then you would need to make the payment directly (which doesn't apply to you since your comment clarified you efiled).