How to transform a Quaternion by a Matrix4x4?

Given a 4x4 matrix, and a Quaternion, what is the most efficient way to compute the matrix-transformed-Quaternion?

What I’m looking for is similar to, but not exactly the following. But I’d like to do it without all these extra operations, and no loss of data (in this example, rotation around the X axis (Vector3.right) could be lost.)

 Quaternion TransformQ(Matrix4x4 matrixTransformation, Quaternion inputQ)
         {
             Vector3 transformedRight = matrixTransformation.MultiplyPoint3x4(inputQ * Vector3.right);
             Quaternion finalQ = Quaternion.FromToRotation(Vector3.right, transformedRight);
             return finalQ;
         }

I would think this can be done by multiplying the Quaternion x,y,z,w components by the appropriate combination of matrix row/column values; I just don’t know the matrix math well enough.

Edit/Update: I finally found this question, which this one is similar to: http://answers.unity3d.com/questions/592099/how-to-muliply-a-quaternion-with-a-matrix4x4.html Alas, this answer yields a result that is a Matrix, and I'm looking for a quaternion.

I also found http://answers.unity3d.com/questions/402280/how-to-decompose-a-trs-matrix.html and am currently using the following base on it, but I'm not sure it's right: is it ok to just ignore the other elements of the transformation?

 Quaternion matrixRotation = Quaternion.LookRotation(
      matrix.GetColumn(2),
      matrix.GetColumn(1)     );
 Quaternion outputQ = matrixRotation * inputQ;