The polarization state generator (PSG) with ergodicity of polarization states (EPS) is an important instrument for measurement of polarization state, and the descriptions of polarization state and polarization devices are the base for control algorithm. A new description of PSG using the quaternion method is proposed and analyzed. A quaternion of three polarization controller is derived, and the equations of rotation axis and rotation angle are obtained. In the experiment, the quaternions of single-stage polarization controller are measured, and the corresponding equations of stress quaternions are obtained when the single-mode fiber is squeezed and the voltage of piezoelectric transducer (PZT) is changed. Finally, using these quaternions, the polarization state generator can be achieved.