You can use the Knife Tool to cut new edges between the verts. Just click K, while in Edit mode, and then select one of the Vertices. After that, just select the one on the opposite side. Now, a new cut will be done. You can finish it now by pressing Enter and then repeat it again, but you can also press E to start another cut, while still being in Cutting mode. When you finish with the grid, just press Enter to confirm and you are done.