The widespread use of new power electronic devices and the integration of renewable energy sources have increased the non-linearity and non-stationarity of modern electric power systems. Taking into account and modeling these nonlinear effects with sufficient accuracy requires the use of quadratic and higher order approximations of nonlinear power system models. However, sufficiently accurate bilinear models of synchronous machine are not easy to find in the literature. Therefore, in this paper we develop a quadratic approximation for the conventional sixth-order model of a synchronous machine using the Carleman bilinearization procedure. The developed bilinear model for one generator can further serve as a basic building block for constructing nonlinear approximations for general power system models.