Did you get a paper check from a Green Dot bank?
Did you have any TurboTax fees that you chose to pay out of your Federal refund? If so, that pay-out-of-refund method is handled by a company SBTPG (aka TPG or Tax Products Group). SBTPG is a Green Dot company.
The following is a scenario in which you might get a paper check from a Green Dot bank. I don't know if this is what happened to you, but it could explain it.
If you chose to pay your TurboTax fees out of your Federal refund, SBTPG's bank would normally receive your refund from the IRS, take out the TurboTax fees, then try to send the rest of the refund to your bank account (or card). If for some reason, that second direct deposit failed, the funds would go back to SBTPG's bank, who would then issue you a paper check in the mail.
If you DID use that method to pay your TurboTax product fees, you may be able to learn some info at SBTPG's refund lookup tool at their website.
To use their refund lookup tool choose the "For taxpayers" portal at:
https://www.sbtpg.com/
