A method and an apparatus for configuring a function of monitoring events concerning user equipment (UE) are provided. The method of a home subscriber server (HSS) includes receiving a first request message including monitoring event information from a service capability exposure function (SCEF), setting, if the monitoring event information is set to loss of connectivity, a mobility-related timer of a terminal to a value based on a value of a maximum detection time included in the first request message, and transmitting a second request message including the monitoring event information and the set mobility-related timer of the terminal to a mobility management entity.