I’m not sure exactly what you mean. When I subdivide a plane or a cube I don’t get any triangles. What exactly do you do when trying to subdivide the mesh?
Subdivided Cube:
Subdivided Plane:
Edit:
By the way, If you want to only split the mesh in two halves, you should use the loop cut tool (Ctrl+R) rather than subdividing.
That has to do with selection of vertices prior to subdividing - I know that when I used to have a single edge selected, I found that the subdivision worked only on the edge selected and caused triangles to flow back to the other verts in the affected faces. Selecting the entire plane and then W, subdivide, should split it evenly in quads then.