First, you need to be sure your state return was successfully filed. If you efiled, be sure your efiled state return was accepted. If you used Online TurboTax, you can check by looking at the Tax Home in your Online account. Does it show there as accepted, rejected, printed, started, ready to mail, or what?
Or did you choose to file by mail instead? TurboTax does not mail it for you. If you chose that method, you have to print, sign, date, and mail it.
The FAQ below tells how to check on a state refund and has a link for your state. It may also have a link to your state's department of revenue. Your state may also have a FAQ section at it's "Where's My Refund" site that tells how long to expect in your state, and how to contact them to inquire.
Choose your state from the table in this FAQ:
FAQ: How do I track my state refund?
https://ttlc.intuit.com/turbotax-support/en-us/help-article/tax-refund/track-state-refund/L3jgO8PGs_...