If I filed a tax extension and paid amount due, do I have to send estimated tax payments?

Paying with a 2021 extension is for the 2021 tax return only.   If you need to file quarterly estimated payments for the 2022 tax year then that is a totally separate situation.

The following is the general rule:

You must pay estimated tax for 2022 if both of the following apply.

1. You expect to owe at least $1,000 in tax for 2022 after subtracting your withholding and tax credits.
2. You expect your withholding and tax credits to be less than the smaller of: 
   1. 90% of the tax to be shown on your 2022 tax return, or
   2. 100% of the tax shown on your 2021 tax return. Your 2021 tax return must cover all 12 months.

Note: If your AGI for 2021 was more than $150,000 ($75,000 if your filing status for 2022 is married filing a separate return), substitute 110% for 100% in (2b)

"I filed a tax extension"

Any 2021 tax was due on April 18th, 2022.

@seawitch2