The Space-based multi-band astronomical Variable Objects Monitor (SVOM) is a collaborative satellite developed by China and France, specifically designed for observing and studying Gamma-Ray Bursts (GRBs) as well as other variable sources. Among its four on-board payloads, the Gamma-Ray Monitor (GRM) is responsible for detecting high-energy photons ranging from 15 keV to 5 MeV, equipped with real-time triggering and localization capabilities. In this paper, we primarily focus on investigating the triggering performance of GRM. Firstly, the energy response matrix of each detector is obtained by using the Geant4 simulation toolkit. Based on the results of background simulations and given samples of GRB, the instrument’s sensitivity and the detection efficiency to GRBs from different directions are estimated. The results demonstrate that GRM exhibits superior sensitivity to GRBs with harder energy spectrum, enabling more than \(80\%\) of the GRBs to be triggered within its field of view. By considering satellite orbit and attitude, we conduct a 3-year simulation of GRB observations which reveals that approximately 106 GRBs can be detected annually in the energy range of 50-300 keV by GRM. Moreover, it is observed that optimal triggering energy range correlates with the hardness index values of the GRBs. Finally, we discuss the on-orbit triggering algorithm that has been implemented by GRM along with developing a ground-based multi-timescale search algorithm for identifying potential GRB events. Our work contributes to understanding the on-orbit triggering performance characteristics demonstrated by GRM, while also providing a benchmark for refining ground-based strategies focused on detecting new instances of GRBs, thus amplifying the scientific output obtained from utilizing GRM’s capabilities.