You do not need to pay the 4th quarter estimated taxes. You will file a final tax return for tax year 2020. If the estimated tax payments do not cover the taxes due for the year (through Oct 2020) then he will owe a tax on his tax return.
Of note, your father will most likely be eligible for a Recovery Rebate Credit on his 2020 tax return in the amount of $600 to reconcile the second round of stimulus payments. Individuals who died after 1 January 2020 are eligible for this credit.
TurboTax will make this calculation for you based on answers and entries when completing the return.
Here is a link with more information on the Recovery Rebate Credit.
For the Fidelity account, there will be no notification for not taking a distribution. If your father had taken a distribution both he and the IRS would receive a Form 1099-R. That Form 1099-R would be used to enter the distribution on his final 2020 tax return.
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