The numerical integration of the non-stationary two-dimensional Schrödinger equation was carried out. In the context of quantum wave-packets, dynamical means, Fourier spectra, uncertainty relations, autocorrelators there were studied the dynamical properties for two coupled quantum Duffing oscillators. It is shown that synchronization for identical oscillatorsand partial transmission of the frequency spectrum in non-identical oscillators takes place. It was found that increasing of noise power leads to the amplitude reduction of the complex-shaped oscillations and the expansion of the frequency spectra.