Quaternion Rotation Around a Given Axis/Vector/Quaternion

I have a question on the usage of the quaternion output.

I see a lot of posts on the net regarding the basic rotations like:
qr = q1 q2 q1^-1
I also see that the rotation between 2 quaternions can be found:
gr = q1^-1 q2

This does not work for me because it gives me a minimal rotation around some random axis. My axis is fixed. And the sensor can be positioned a bit off the axis.

My problem is this:

I need to measure a rotation angle of my sensor around the vector/quaternion which I know.

So what I have is q1 & q2 are subsequent readings of the sensor.
What I need to find - a rotation of this sensor around the axis, defined by q0.

Any help with this would be greatly appreciated.

Thank you!