Do I need to make quarterly tax payments in 2021 for the Capital Gains I will owe from a stock sale in May 2021 that I will have to claim on my tax return in 2022?

[Critter-3]

Possibly ... 

Have you tried using the IRS withholding estimation tool:

https://www.irs.gov/individuals/tax-withholding-estimator

Or have you tried using TurboTax's Tax caster tool:

https://turbotax.intuit.com/tax-tools/calculators/taxcaster/

If you don't pay the estimates and your withholding isn't any of the following you will be hit with underpayment penalties.

1) 2020 balance due after withholding is less than $1000

2) withholding covers 90% of 2020 tax

3) withholding is 100% (110% if 2019 adjusted gross income is over $150,000) of 2019 tax

See this for calculating 2020 estimated taxes - https://ttlc.intuit.com/community/tax-payments/help/can-turbotax-calculate-next-year-s-federal-estim...

See this for paying estimated taxes - https://ttlc.intuit.com/community/tax-payments/help/how-do-i-make-estimated-tax-payments/00/25875

[fanfare]

if you take money from your 401k   the custodian will charge 20%. They don't do any estimating for you.

It's the same with stock profits. But you have more control.

OR

take your chances and hope the penalty is small or zero.