make estimated tax payment by the quarterly due date for the quarter in which taxable event occurred.

otherwise you risk underwithholding penalty.

 in some cases, your circumstances will avoid the penalty.

you need to do the worksheet if it is of concern to you.

Q. When is  tax on a Roth IRA conversion due?

A.  As Fidelity said, it's "due" April 15 of the following year.  The Roth conversion is not taxed, in isolation.  It is added to your other income to determine your total tax liability for the year. 

However, if the distribution is large enough to result in a significant tax bill, you may be subject to an underpayment penalty.  So you you may want to make a quarterly payment for the quarter you make the conversion. Quarterly payments (for 2020) are due July 15 (1st & 2nd Qtr), Sep 15 and Jan 15, 2021. 

You should make estimated tax payments for the current tax year if both of the following apply:
- 1. You expect to owe at least $1,000 in tax for the current tax year, after subtracting your withholding and credits. 
- 2. You expect your withholding and credits to be less than the smaller of: 90% of the tax to be shown on your current year’s tax return, or  100% of the tax shown on your prior year’s tax return. (Your prior year tax return must cover all 12 months.)

Alternatively, you might be able to increase your W2 withholding, at your regular job, to account for the extra income