The hypothesis of a PeVatron in the Galactic Center, emerged with the recent $γ $ γ -ray measurements of H.E.S.S. [1], motivates the search for neutrinos from this source. The effect of $γ $ γ -ray absorption is studied: at the energies currently probed, the known background radiation fields lead to small effects, whereas it is not possible to exclude large effects due to new IR radiation fields near the very center. Precise upper limits on neutrino fluxes are derived and the underlying hypotheses are discussed. The expected number of events for ANTARES, IceCube and KM3NeT, based on the H.E.S.S. measurements, are calculated. It is shown that km$^3$ 3 -class telescopes in the Northern hemisphere have the potential of observing high-energy neutrinos from this important astronomical object and can check the existence of a hadronic PeV galactic accelerator.