a surface is 2D, yes, it can be in any 3D shape but it has length and bredth dimensions only (ie 2D), so you can only give it 2D (ie thin) elements.
if you want to fill the volumen enclosed by a suface, now you're talking a 3D shape, and you have to use 3D (ie solid) elements.
it's a different question if you're asking tri or quad. Generally quads are better than tri, they suit the math better. but quads can have difficulty mapping surfaces (4 points don't make a flat plane); tris can mapp any surface (3 points define a flat surface) but their results can be "suspect".