You have to distinguish between modeling and later useage of the final result. Game models are usually modeled with quads first. Edgeloop tools for example doesn't work at a tri geometry. But then exported with a triangulated geometry to have the geometry as stable as possible.
Some game engines imports quads too. And when it's static geometry then it doesn't matter. But game engines doesn't necessarily import the quads with the same vertices order than what you had in Blender. And then the faces may bend into a wrong direction. So you usually triangulate it before export, and then check if all edges really points in the desired direction for your animation needs.
When it's a game model anyways, then you can also model with tris already. When it makes sense. As told, some tools works better with a quad geometry.