How would record the monthly expense of the quarterly payment? Say your quarterly payment is $600, how could you record the accrued expense of $200 per month?
In Quicken - you can create a Manual Reminder with the special tax due dates - scroll and select the —> Pay Estimate Taxes -
This sounds more like a budget question, and also not even about taxes which is what this thread is about.
The reason I said that it might not be about taxes is because even though the government calls these "quarterly" in reality it is the most mixed up "quarterly" you can imagine. 1/15 (which is for the prior year), 4/15, 6/15, 9/15. There isn't going to be a perfect monthly amount that works for this because they aren't evenly spaced.
If you actually have a real quarterly expense, then you would be using Quarterly from the reminder selection that @Ps56k2 pointed out.
But if you want to budget for it, then you would put in a monthly budget of 1/3 of that and use a rollover.
To budget for taxes monthly you would have to both use the rollover and also set different budget amounts for the different months. For instance, between 4/15 and 6/15 the monthly amount would have to be larger than 6/15 to 9/15.