Include quaternion chirality in code?

I have quaternion data that I am trying to switch from the right Y-up coordinate space to the left Z-up coordinate space. It follows what I need to do switch the chirality.

The only information I can find is this

And I can’t wrap my head around me.

Say I have my Quaternion like:

public static float[] quat= new float[4](0.70711,0.70711,0,0);

In C # code, how do I switch chirality?

I tried to negate the axis and simply replaced the values ​​of Zand Y. I tried to multiply the quaternion by another quaternion equal to 90 degrees, as suggested in this link. and still see irregular rotations.

Thank.