We study the quantum dynamics of a two-electronic state model system with a conical intersection involving two vibrational degrees of freedom, x and y. The two diabatic surfaces are displaced only along x, while the interstate coupling is linear in y and contains Gaussian damping factors along x and y. The role of the initial momentum and of the difference of the slopes at the conical intersection is discussed. In particular, the analysis of the numerical results shows that the diabatic transition at the conical intersection is accompanied mainly by a massive flow of vibrational quanta from the x to the y oscillator of the same electronic state. Furthermore, we propose a simplified time-dependent monodimensional model in which the x oscillator is assumed to move as an independent classical oscillator, while the y oscillator is described quantum-mechanically. Despite the crude approximations involved, this simplified model reproduces quite well the exact results for the transition probabilities at the first passage through the conical intersection and candidates itself as a useful tool for treating multidimensional systems with low computational effort.