Sorry, if this is an additional refund of state income tax, it is treated like all other refunds of state tax. It is not deductible if you used the standard deduction for your 2021 income taxes, but it may be taxable if you used itemized deductions and deducted your state income tax. (It's taxable because it is a rebate or refund of something you previously deducted.)
Turbotax can figure this out for you if you modify your program input. Go to the "refund of state income tax" section on the income page. Add the rebate amount to whatever your previous refund was. In other words, Turbotax will normally say something like "last year, you got a California refund of $500, is that correct?" Say no, and add the amount of the rebate to your regular refund. Turbotax will ask some other questions to determine if the refund/rebate is all or partly taxable. (If you did not use Turbotax last year, the program will ask "did you get a state tax refund" and will then ask questions about last years tax return. Add the rebate to the amount of your regular refund.)
As to timing, if you did not physically receive the check or bank deposit until 2023, you are not required to report it until 2023. But, the FTB or the IRS may send you a letter and a tax assessment if they believe this was 2022 income, and you would have to reply with an explanation and proof. It may be simpler to report it for 2022 even though that is not technically correct.
