Typically, after you pay off your mortgage, the lender will file a Release of Deed of Trust with the County Clerk and Recorder's office in your county. Once that's done, so long as this was your only mortgage, you will now own your house outright.
Remember that your mortgage company may have been collecting from your monthly payments certain amounts that they set aside ("escrow") for your homeowner's insurance and annual real property taxes. Thus, you will now want to be sure to pay these directly to your insurance company and County Tax Assessor directly. Usually these are not monthly payments, though the payment schedule with insurance companies can vary from monthly to quarterly to semiannually to annually, so make sure to work that out with your insurer.
Tax payments to the county are usually made once annually, due on April 30. However, you can also choose to pay half of your tax bill by February 29 (this year, anyway) and the other half by June 15.
So the short answer is: No, you probably will not be making any monthly payments. I hope that helps.
This answer is not intended as legal advice. It is generalized information, and no lawyer-client relationship is hereby established.
Once the mortgage is paid in full, you have no further obligation to pay the lender. However, you need to be certain the lender records a Satisfaction of Mortgage promptly. That is the document that provides a public record of your payment.