a quad is essentially made up of two tris with the diagonal edge hidden. At render time(and real-time render) the algorithm will recognise tri faces. However, If the mesh is built initially from quads, it will be a quad mesh and will be considered such when UVing, sculpting, etc…
Most 3D apps(Max,Maya,etc) have tools to check a mesh’s properties. For instance: number of tris/quads/n-gons, rat’s nests(tri-shaped quads) overlapping verts/edges…
It’s very important that you use these tools if you intend to use a mesh in Mudbox as it can be a real headache to work with geometry with bad topology.
You should read the mudhelp files as there is a clear explanation of specs in terms of preparing a mesh for export to Mudbox.
Also, the mesh you have there is useless as it is as a sculpting base and should be scapped or rebuilt.