I have two vectors u and v that both lie on z = 0. Can I find a quaternion axis to for these vectors if the axis must also lie on z?

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Vectors u and v both lie on the x-y plane or have a z value of zero. I understand that there is an infinite amount of quaternions that will rotate u to v. Is there a way to find an axis for a quaternion between these two points that also lies on the x-y plane or has a z value of zero? I've tried multiple ways to do this, however I am fairly new to the concept of quaternions and every axis I have found always deviates from the x-y plane. Is this quaternion even possible?