For a lump sum of income in April, the quarterly payment is due by June 15. You can make a payment directly on the IRS web site at www.irs.gov/payments.
There are two ways to think about the payment. If you want to pay the approximate tax you owe, it will either be 15%, 22%, or probably 24% of the converted amount. Which percentage depends on how much your other income is. Check the brackets, figure out where your regular income puts you, and then figure you will owe tax at the highest rate that applies to you. https://www.bankrate.com/finance/taxes/tax-brackets.aspx
Or, pay the minimum needed to avoid a penalty. This is 110% of your tax liability for 2019. Let's assume your tax liability for 2019 was $9000. That means you must pay into the system at least $9900 for 2020. If you anticipate your job withholding will be $8000, then you would pay $1900 to get to that 110% level for 2020.
Remember that if you pay less than you owe, you will have to pay it when you file next year. If you pay more than you end up owing, you will get the difference back as a refund.
