Here are a couple of options to handle this. My favorite, by far, is #3 once I learned this trick.
#1) If you do the conversion in the fourth quarter you do have the option to only pay the 4th quarterly estimated payment. The problem with this scenario is that your tax return may be more complicated in that, in the general case, you may have to calculate your income and payments for each quarter separately to avoid a penalty. (Kind of like doing 4 tax returns instead of 1 IMHO). Estimated payments are really designed to be 4 equal amounts sent in throughout the year. Otherwise you risk complicating your tax return.
#2) To solve this problem, I think you might be able to pay the entire estimated payment in the first quarter. The disadvantage would be the feds would have your money longer and this may actually have an economic difference with the rising interest/rates and inflation. You would also need to know the amount that needs to be paid via estimated payments early in the year. I would need to verify if this indeed simplifies your return vs option 1 or not.
#3) You could do what I do. Take a distribution from a retirement account (401k, trad IRA, Roth IRA etc.) that allows you to specify tax withholding. I use Vanguard IRAs for this purpose. Instead of the money coming to you, most of it would go the withholding. Then within 60 days do a roll-over with after tax money back into the account. I usually do this in December each tax year to cover all my Roth conversions. You have to remember that you can't do more than one of these every 365 days. The advantage of tax withholding vs estimated tax payments is that any tax withheld is assumed to have come in uniformly over the year. It doesn't matter which day of the year that you do this (assuming that you observe the 365 day rule.) The other advantage is that the 1099-R will capture the amount of withholding so that you don't have to "remember" that you paid estimated taxes.
-gauss