Can z value in quaternion get negative?

edit

If you have a rotation quaternion q=(x,y,z,w), then -q=(-x,-y,-z,-w) describes the same rotation, so you cannot rely on z (or any other component) being positive.

A rotation quaternion is essentially a rotation axis (x,y,z) and an amount of rotation w (I avoid the word angle because it's not an angle in the usual sense), scaled to unit length (|q|=1). If you change the sign of w, you reverse the rotation direction. The same thing happens if you mirror the rotation axis to (-x,-y,-z). So if you do both, these effects cancel each other out.