From what I can remember in making these, you sew the hypotenuse together for the half-square triangle, which would be along the bias of the fabric. the outside of the square would be straight of grain. I have never done a quarter-square triangle, but according to the photo, you would still sew the hypotenuse together, but this would be on the straight of grain instead of the bias.
If you made some half-square triangles, sewed them together and then cut from the center of the hypotenuse to the top, you would end up with two quarter square triangles, sewn into a half square triangle. I believe that the grain would be correct.