How do I report last year's state or local refund without a 1099-G? When I clicked on this question, it said they couldn't find my answer.

If you know last year's refund amount, you don't need a 1099-G. Follow these steps to enter last year’s refund amount:

1. Sign in to your account and open the Wages and Income section of your return.
2. Under Income and Expenses, select Add more income.
3. Look for the Other Common Income section and select Show more.
4. Select Start on the Refunds Received for State/Local Tax Returns section. 
5. Enter your refund amount.

For more information, please also see: Where do I enter a 1099-G for a state or local tax refund?

Thanks.

Unfortunately, I do not know the amount. I am trying to find it. I am not filing jointly this year, and my ex needs these values, and  I do not know them.

Did you file Joint last year?  Do you know if you took the Standard Deduction or if you itemized deductions?  If you took the Standard Deduction then the state refund is not taxable and your ex doesn't need it unless he needs it for something else.  

Did you use Turbo Tax last Year?  You have to log into that account.  First LOG OUT of whatever TurboTax account you're logged into right now. Then use this TurboTax account recovery website to get a list of user ID's for an email address. Run the tool against any email addresses you may have used
https://myturbotax.intuit.com/account-recovery/

If you used the Desktop CD/Download program then the only copy is on your computer and not saved or stored online.

You can also request a transcript from the IRS
https://www.irs.gov/individuals/get-transcript

If you got divorced in 2020 here's some info for you.....

If you used Turbo Tax last year you should start a brand new account for yourself and not transfer from last year.  You can not remove a spouse or switch the order of names.

Online is only good for 1 return.  But you can use the same email address for 5 accounts.   How to start another return in the Online version  

https://ttlc.intuit.com/community/using-turbotax/help/how-do-i-start-another-return-in-turbotax-online/00/25596

Divorced FAQ

https://ttlc.intuit.com/community/family/help/what-if-i-was-divorced/00/25830

Start a new account. Removing a spouse is not recommended

https://ttlc.intuit.com/community/family/help/how-do-i-remove-a-former-spouse-from-my-return/00/27202

Getting Divorced article

https://turbotax.intuit.com/tax-tips/marriage/getting-divorced/L20NC66cf

Video tips for Divorced couples

https://turbotax.intuit.com/tax-tools/tax-tips/Family/Video--Tax-Tips-for-Divorced-Couples/INF14631.html

And see what AGI to use

https://ttlc.intuit.com/community/agi/help/what-agi-should-i-enter-if-i-m-recently-divorced-or-widowed/00/26954