Each quad face is always made of 2 triangles in 3D softwares. I guess the result you see has to do with the way Blender will triangulate each of the 4 faces it creates when it subdivides.
Here I use a grid as image texture to make it more understandable. On the left, the original object, on the right, subdivided once. Here is how Blender triangulates:
If I ...
You can bake this. Give the texture mesh an emission material. Select the object, then the target object (SHIFT).
In an image editor create a new texture (square, left)
Add an image texture node to the material of the target object,
select your new texture. It does not have to be connected to
anything, but make sure it's active (clicked on)
In the render ...