No...you don't need to do that.
State taxes that were not withheld or paid during the 2020 calendar year (1Jan to 31 Dec) in the first place, cannot/are not to be used as a deduction in your Federal tax return for 2020.
So you will just prepare and file as you normally do....BUT...any taxes that you pay to your state when you file 2020 state taxes now (during 2021)....that state balance due payment can be used next spring as a 2021 Federal income deduction, along with any state withholding you may start doing now.
All this assumes you have enough total deductions such that Itemized Deductions can be used.
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SS Admin doesn't do state withholding, but your pension/401k/IRA administrators usually can/will allow it to be set up.
____________*Answers are correct to the best of my knowledge when posted, but should not be considered to be legal or official tax advice.*
