You can manipulate quaternions in BJS. Have you seen Quaternion - Babylon.js Documentation
Just to emphasise something a little different between terminolgy in the video you gave the link to and BJS
In BJS (and Wikipedia) quaternion has the form a + x i + y j +z k and not all quaternions are rotation quaternions.
A rotation quaternion should be a unit quaternion In the video it appears the term quaternion is restricted to one having a unit vector for its axis.
Here is an example with the quaternion randomized and normalized to produce a unit quaternion