No, you don't. All current-generation commodity GPUs use (and have used for some time) triangle-based rasterization methods exclusively. Even though older version of OpenGL support the
GL_QUADS rendering mode, these were converted to triangles by commodity GPUs. It's likely that
GL_QUADS only resulted in actual quadrilaterial-based rasterization on esoteric academic hardware or hardware used for high-end offline 3D rendering in the early 80s or 90s (I have no evidence to support this claim, I'm just postulating).
That said, looking at quadrilateral (and in general, polygon rasterization other than triangle rasterization) can still be educational, and thus useful, simply by providing a different perspective of things. The algorithms involved in doing so efficiently are interesting and sometimes still have applications beyond graphics programming. I once used a edge-walking polygon rasterization technique as a means of implementing a simplistic fake 2D water simulation, for example.
It also may help you understand why triangle rasterization is preferable, in part due to the potentially non-planar nature of polygons other than triangles and because of some of the resulting optimizations you can make with triangles.