Transform

class Transform
{
public:
  calcMatrix(); // builds a matrix from m_T and m_R
  setFromMatrix(const Matrix& mat); // sets m_T and m_R

  getRotation();
  setRotation(const Quat& r);

  getTranslation();
  setTranslation(const Vect3& t);

  rotate(const Quat& r);
  translate(const Vect3& t);

protected:
  Quat m_R;
  Vect3 m_T
}

You don't have a quat transfrom. Quat can only describe rotation.

You have a rotation and a translation (and if your brave, a scale), which are combined together to make a matrix.