You make one payment to the IRS for quarterly estimated taxes. As part of your actual tax return for the year, you will list your income and expenses, calculate your taxable profit, your income tax owed, and include a schedule SE for the self-employment tax. If you under-paid your estimate, you will owe more tax, and if you overpaid your estimate, you will get a refund.
The IRS will use the information from schedule SE to determine how much credit you get with the social security administration for the social security part of your self-employment. You can check your online account with the social security administration to make sure everything is recorded correctly, although it may take a few months after you file your return for everything to be updated.
In other words, you make one estimated payment, and the IRS will divide it up based on your schedule SE.
