I would like to know how to record a refund check from unemployment compensation fund?

I know how to record the refund check in QuickBooks Online, SUEP651.

You'll have to make a deposit. Then the source should then be the expense account for taxes paid. The process helps reflect the amount from your bank. I'll guide you on how:

1. Click the +New button and select Bank deposit.
   
2. Choose the bank account.
3. In the Add funds to this deposit section, enter the required fields.
4. Select Save and close.
   

If you're unsure what expense account to use, I recommend consulting with your accountant so they can provide you with the appropriate one. This is to ensure your books are accounted for properly.

I'm also adding this article for more details about recording a bank deposit in QBO: Record and make deposits in QuickBooks Online.

Get back to me if you have other questions about managing your account. I'll be around to help.