Nonlinear Landau-Zener tunneling in two-dimensional photonic lattices of Kerr nonlinearity media is investigated. We first derive the nonlinear three-level Landau-Zener model in photonic lattices for the case of hexagonal symmetry, and then simplify this model to a nonlinear two-level Landau-Zener model under a certain initial condition. For the three-level tunneling model, we calculate the tunneling probability numerically and study its changing rules under the specific initial condition.