No. I do my MIL’s taxes and do this for her every year. There is no maximum. They allow quarterly payments just because most people can’t manage that big a cash payment up front.
I can’t dispute this as a generalization but government agencies are usually very conscious of cash flow and cash those checks ASAP. That has been my experience with the states I’ve lived in, as well as any business that has automated payment intake systems.
Or self-employed.
I’m a retired federal employee, and I have Federal tax withheld from my pension payments so I don’t have to make quarterly estimated tax payments. If I had to pay state income tax, I could also have that deducted from my pension payments.
The general idea is that you should be paying tax fairly soon after you receive the income, so that you are not treated significantly more favorably than earners who have withholding deducted.
But you’re not expected to pay tax in advance on income that you haven’t yet received.