I am rotating a unit vector around the z-axis. I need to calculate the tangent at each point and draw a quad with the extends of the quad along the tangent. This is essentially like rotating a point around a sphere of radius 1.0 with center at 0,0,0.

I believe I have found one tangent so I can draw one axis of the quad. This error leads me to believe I have to find two tangents.

A---B
|
|
C---D

Where the center of the quad is the point I'm finding the tangent on. Each point A,B,C,D is the extends of the quad along the tangents.

Not even sure if this is the best approach. Essentially I need to draw a quad at each point along a sphere.

I'm not sure I'm getting what your asking for, but here's a try...

Where V is one of the unit vectors that you speak of, and Z = (0, 0, 1) is the axis of rotation that you speak of, then two orthogonal unit tangent vectors are simply Z and Z x V, where x is the cross product operation. Does this help get you started, or did you already have this?

Getting the four corners of the quad from these two orthogonal unit tangent vectors seemingly requires some averaging and some negation ops here and there, maybe some scaling / normalization if you please, and likely nothing more complicated.

Also, for what it's worth, a circle is a 1-sphere.