I have a 1099-INT Interest income for a personal checking/savings account in NJ and do not know if I can select 'tax exempt in NJ/taxable federal' or not?

You'll need to check on the website of your NJ state tax board.  Do a search on their website for "personal interest income".  I have an idea you will find personal checking/savings interest is taxed as regular income.  However, if you were to invest in in-state (NJ) municipal bonds, very likely the interest made on the bond(s) would be tax-exempt in your state return.

Yes, bootjan is correct. Typically, however, interest from personal checking/savings accounts are not tax exempt.

For your reference, 2019 New Jersey Instructions for Form NJ-1040