Spontaneous synchronization is a significant collective behavior of weakly
coupled systems. Due to their inherent nonlinear nature, optomechanical systems
can exhibit self-sustained oscillations which can be exploited for
synchronizing different mechanical resonators. In this paper, we explore the
synchronization dynamics of two membranes coupled to a common optical field
within a cavity, and pumped with a strong blue-detuned laser drive. We focus on
the system quantum dynamics in the parameter regime corresponding to
synchronization of the classical motion of the two membranes. With an
appropriate definition of the phase difference operator for the resonators, we
study synchronization in the quantum case through the covariance matrix
formalism. We find that for sufficiently large driving, quantum synchronization
is robust with respect to quantum fluctuations and to thermal noise up to not
too large temperatures. Under synchronization, the two membranes are never
entangled, while quantum discord behaves similarly to quantum synchronization,
that is, it is larger when the variance of the phase difference is smaller
