Due to abundant dark matter and stellar-rich environments, the galactic center is the optimal location to probe for signatures of captured dark matter annihilation. In this work, we have studied the dark matter capture in the galactic center neutron star distribution and analyzed the neutrinos coming from the annihilation of captured dark matter through long-lived mediators in an idealized gigaton neutrino detector like IceCube/KM3NeT. We report projected limits on the dark matter-nucleon scattering cross-section that are orders of magnitude below the current limits in the TeV-PeV mass scales.