Can the state take your income tax check?

Yes this could be very possible. The Department of Treasury's Financial Management Service (FMS), which issues IRS tax refunds, has been authorized by Congress to conduct the Treasury Offset Program. Through this program, your refund or overpayment may be reduced by FMS and offset to pay any past due child support, Federal agency non tax debts, or state income tax obligations.

Go to the IRS gov website and use the search box for Topic 203 - Failure to Pay Child Support, Federal Non Tax and State Income Tax Obligations

For additional information, FMS can be reached at 800-304-3107.