Here's an answer copied from
Estimated taxes used to be paid based on a calendar quarter, but in
the 60's the Oct due date was moved back to Sept to pull the third
quarter cash receipts into the previous federal budget year which
begins on Oct 1 every year, allowing the federal government to begin
the year with a current influx of cash. That left an extra month that
had to be accounted for in the schedule somewhere. Since individuals
and most businesses report taxes on a calendar year, the fourth
quarter needed to continue to end on Dec 31 which meant the Jan 15 due
date could not be changed, that left April and July 15 dues dates that
could change. April 15 was already widely known as the tax deadline,
so the logical choice was the second quarter which had its due date
changed from July 15 to June 15.
The explanation for the Sept. payment makes sense. The rest doesn't. My favorite suspect would be the federal budgeting process. I think, that the standard process is to have a budget in Congress for debate by the beginning of Q3 (6 months to get next budget approved). Executive and Congress would likely prefer to have the Q2 money in the bank, rather than forecasted for a month in the future. They would have to estimate 5 quarters of income instead of 6 when they debate the budget.