Your post is not clear. If you owed taxes to the state, then you will not get a refund from the state. Or are you saying something else?
You can look on your state tax return that was filed to see whether you owed taxes or were getting a refund.
The state and federal are totally separate. If you are referring to a refund from the federal even though you owed state, you can track the federal refund at the IRS tracking web site: https://www.irs.gov/refunds