The axion-like particles (ALPs) are very good candidates of the cosmological dark matter, which can exist in many extensions of the standard model (SM). The mass of the ALPs can be as small as O ( 10 - 22 ) eV . On the other hand, the neutrinos are found to be massive and the SM must be extended to explain the sub-eV neutrino masses. It becomes very interesting to consider an exclusive coupling between these two low scale frontiers that are both beyond the SM. The propagation of neutrinos inside the Milky Way would undergo the coherent forward scattering effect with the ALP background, and the neutrino oscillation behavior can be modified by the ALP-induced potential. Assuming a derivative coupling between the ALP and the three generations of active neutrinos, possible impacts on the neutrino oscillation experiments have been explored in this paper. In particular, we have numerically studied the sensitivity of the deep underground neutrino experiment (DUNE). The astrophysical consequences of such coupling have also been investigated systematically.