Simulation of neutron transport process plays an important role in nuclear reactor computation and the numerical technique becomes the focus of nuclear reactor engineering. This paper provides a neutron finite volume lattice Boltzmann method (NFV-LBM) for solving the neutron discrete velocity Boltzmann equation (NDVBE), in which the NDVBE is deduced from the neutron transport equation (NTE) and the NFV-LBM is obtained by integrating the NDVBE. The macroscopic conservation equations recovered from the NDVBE via multi-scale expansion shows that the NDVBE has higher-order accuracy than diffusion theory, and the numerical solutions of neutron transport problems reveal the flexibility and applicability of NFV-LBM. This paper may provide some alternative perspectives for solving the NTE and some new ideas for researching the relationship between the NTE and other approximations.