Triangles in 3D euclidean space have the advantage of being uniquely defined by 3 vertices that have the nice property of always sitting on a single plane. There's no linear transformation you could apply that results in something other than a well-defined, planar triangle. The worst that might happen is collapsing it into a line or a point.

The same cannot be said for quads. If you e.g. have 4 vertices where 3 are sitting on the same plane, but the 4th isn't, there are 2 different, possible ways to subdivide this into triangles. In addition to all the possible non-linear surfaces one might image occupying that space.

Old OpenGL and Direct3D versions did have quad and arbitrary polygon rendering support. IIRC from vague memory, the above scenario was something you were supposed to avoid and results varied between graphics card vendors.

On a side note: If we are talking purely 2D, some old 2D software rendering systems used trapezoids as basic primitives, defined by their top and bottom edges. For instance, the X11 XRender API explicitly supports drawing trapezoids. They are easy to rasterize, interpolate across and quite flexible. Many other 2D shapes can be conveniently composed from them, including screen space triangles, if you were to implement a software rasterizer.