Schedule H and Estimated Taxes

No.

If payroll is submitting your portion of the FUTA tax and then recouping that from you via quarterly payments you make to them, there would be no additional reporting necessary on your part. If you are not sending quarterly estimated tax payments directly to the IRS, the IRS would not have a record of estimated payments. 

You need to know what the Payroll Service is doing.

I am reading that you are sending a quarterly payment TO YOUR PAYROLL SERVICE not to the IRS.

Does the payroll service then use these funds to reconcile the Schedule H and file everything for you?

If this is the case, you would not claim estimated payments on your 1040 and you would also not file Schedule H since the payroll service did all this for you.

Schedule H computes your tax liability for the employee taxes (FUTA) relating to the hiring/paying of a household employee and it sounds like your payroll service is submitting that for you. 

HOWEVER- If payroll is only computing and paying the employee's share, then you would need to file Schedule H and pay your share through your 1040. 

Again, ask the payroll service what they are doing, what is included in the payments you make to them, and what reporting responsibility (if any) you have.