We explore experimentally the effect of nonlinearity on resonant coupling between high-symmetry momentum states in hexagonal photonic lattices. We observe nonlinear Pendellösung oscillations with the power-dependent excitation of additional critical points of the same symmetry in the reciprocal cells closest to the excitation point. In contrast, the nonlinear Landau-Zener tunneling in biased lattices exhibits a sharp transition to the modulational instability with an increase of the input optical power.