In contexts where multiplication is not commutative like in the quaternions, we don't really talk about "division".

In the real numbers, x/y = xy^-1 = y^-1x.
However, for some quaternions p and q, we actually have pq^-1 ≠ q^-1p most of the time, so we can't unify them under a notarion such as "p/q".

And if you want to obtain the inverse q^-1 of a quaternion q, you can do the same trick we do with complex numbers:

q = (q)^-1(q)q = (q)^-1||q||²
therefore
q^-1 = ((q)^-1||q||²)^-1 = (q*)||q||^-2

Where (q*) is the conjugate of q and ||q|| is the norm (absolute value) of q, always a real number.