It sounds as if the state has changed something on your state return and instead of getting a refund you now owe. Only the state can explain. TurboTax does not get any information from your state after you file your tax return. How were you informed that you owe the state? Did they send you a letter? What does it say?
TurboTax does not receive any information from the IRS or your state after your return was filed, so no one at TurboTax knows about a letter you received from the IRS or the state.
TurboTax has a special phone number for help with IRS/state tax letters, which will be easier than going through regular Customer Support. At the page below, click on the blue button that says "Get Help from TurboTax Support". Then it will ask you what tax year is your letter. Then it will ask you what the letter concerns. Then it will show you the phone number during posted business hours.
https://support.turbotax.intuit.com/irs-notice/audit-support/