back door roth IRA conversion and payment of tax-quarterly or not

If you're doing a true back door Roth it means you are beyond the income threshold for receiving a tax deduction on your traditional IRA and you're also phased out for income purposes regarding a Roth IRA.  This means the money going into the IRA should be after tax dollars which will then be transferred into a Roth.  You will receive a 1099R to be used for filing on your next tax return where no additional taxes should be paid.

You don't pay tax on a "back door" Roth IRA conversion.

If you are doing a regular Roth IRA conversion, the taxes are due in the quarter the income was realized.