How to find change in angle when vertical Scale of depth axis is modified?

Question

Let Q be the Quaternion that perform a rotation of angle degrees around some axis of rotation. Later I change vertical exaggeration of my Vertical axis (z) to some value VS, e.g. make the vertical (z) scale to 50% or 25% etc without altering the z-values/range? How does it effect angle (w) and orientation(x,y,z) of Quaternion Q. How do I find this change in angle.