For the simplicity, we'll consider two 3D points, that moves one relatively to other, in time. Let's say:

at moment t0, we have P1(0,0,0) and P2(0,2,0)
at moment t1, P1 is still (0,0,0) but P2 changed to (0,2,2).

From what I've understood reading about quaternions, is that, at moment t0, Q1 (representing P1) and Q2 (representing P2) will be both (0, 0, 0, 0).

But at the moment t1, Q2 will become something else (w, x, y, z). How do I calculate the Q2 at t1 moment?

I've googled a lot on this subject, but I was able to find only rotation between quaternions. I will appreciate any guidance. Thanks!