Here we develop the nonequilibrium Green's function formalism to analyze the fluctuation spectra of the boson tunneling currents. The approach allows us to calculate the noise spectra in both equilibrium and nonequilibrium conditions. The proposed general formalism is applied to several important realizations of boson transport, including the tunneling transport between two reservoirs and the case where the boson current flows through the intermediate region between the reservoirs. Developed theory can be applied for the analysis of the current noise in waveguides, coupled optical resonators, quantum microcavities, etc., where the tunneling of photons, exciton-polaritons, or excitons can be realized.