If you have proof like your bank statement or credit card charge send them a copy. You can't pay a tax due from Turbo Tax.
You probably just paid the Turbo Tax fees for doing the state return. And you can not have a state tax due deducted from a federal refund, only the Turbo Tax fees. Turbo Tax just shows you a screen at the end showing the Net amount of federal and state combined. But it is just for your own info. You do still have to pay the state yourself separately.
Your only proof of payment is your bank statement or credit card showing the payment coming out. Turbo Tax only passes your bank account number to the state for the state to take it out of your account. But most states do not let you pay by Direct Debit so you have to mail in a check or go directly to the state's website to pay.
Check the printout or PDF of your return; look for the state cover sheet with the Turbotax 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/questions/1900884-what-are-my-options-for-getting-my-state-refund-or-for-pay...
And some banks use different routing numbers for deposits and payments so double check the debit info with your bank.
