Once a tax return has been Accepted by the IRS or a State, TurboTax receives no further information concerning the tax return or any tax refund.
Did you e-file your state tax return and was it accepted?
You complete your tax return by finishing all 3 Steps in the File section. In Step 3, to e-file your tax return, you must click on the large Orange button labeled "Transmit my returns now".
After completing the File section and e-filing your tax return you will receive two emails from TurboTax. The first email when your tax return was transmitted and the second email when the tax return has either been accepted or rejected.
Use this TurboTax website to check the status of an e-filed return and if it was either accepted or rejected - https://turbotax.intuit.com/tax-tools/efile-status-lookup/
If accepted by the state use this TurboTax support FAQ to check the New Mexico state tax refund status - https://ttlc.intuit.com/questions/2896789-how-do-i-track-my-new-mexico-tax-refund