Blender works only with quads and triangles, which is to say faces with four vertices. If you subdivide a cube, you subdivide it from 6 quads into 24 quads. Each “edge” of the cube is actually two edges. You can’t make one side a single face with the other sides being subdivided because that one face would have 8 vertices. Does this make sense?
If not, I may be misunderstanding what you’re trying to do. It might be best if you include an image making it clear what you’re after.
The face of the aforementioned cube is only subdivided one-way, so there are no vertices in the middle of it. I want to remove the edge that subdivides the face.
If this is really all you want, and there are no other faces involved, or if the edge you want to remove is part of a continuous loop you want to remove, then alt-right click on the edge (this will select the whole loop, if there is one) press X and select Erase Edge Loop.