As long as the a triangle is defined by three non-colinear vertices (read: none of the angles are exactly Pi), then the vertices define a unique plane.
A quad is, of course defined by four vertices. It's perfectly possible for those vertices to be non-coplanar. In that case, your quad would really be two triangles divided by a diagonal on the quad. That's two planes, two sets of surface normals, etc.
Every available modeling tool, every algorithm for texturing, lighting, etc all assume that a model is made of plane segments, and every formula we have (cross products for normal calculation is the first one we have) use the absolute minimum input dataset - three vertices define a plane, and the plane is what we need to do all of the fancy stuff.
You could certainly write an engine to work with quads, but you'd find yourself ignoring the forth vertex in just about every case, except for when you (frequently) would need to make sure it's coplanar with the other three that define the quad. And, the most logical solution to the case where it is not coplanar would be to split the quad into two triangles. So, why not just do that to start with?
What on earth would be the point of working with quads?
If you want a quad, put two triangles together.