Rotate an object from an axis to another axis

Hello, I have two objects A and B with two differents coordinates sytems (Object A the Y axis is up and for object B the z axis is Up).

I would like to rotate the object B to have its x axis align with the x axis of the object A.

I tried with quaternion but all the object B is rotating to match with the A. I would like the object B rotates around its Z axis until the two x axis are align.

Thank you in advance.

 

For more explanations :
I have :



I want :



The only way I found is to make
objectB.rotation = ObjectA.rotation
and add the right Euler Angle to rotate the trasform of the B Object.
But if there are a solution to directly align the axis it will be better