You are not required to pay any estimated taxes (the IRS will not send any letters about missing or short estimate payments - it is your responsibility to keep track of your taxes). But if you don't withhold or pay enough tax on income received during each quarter, you might get hit with an underpayment penalty.
Underpayment penalties are assessed if you don't withhold or pay enough tax on income received during each quarter. In fact, it's entirely possible to get hit with an underpayment penalty even though you paid your tax bill in full by the April deadline or are getting a refund.