  We investigate neutrino oscillation in the field of an axially symmetric
space-time, employing the so-called $q$-metric, in the context of general
relativity. Following the standard approach, we compute the phase shift
invoking the weak and strong field limits and small deformation. To do so, we
consider neutron stars, white dwarfs and supernovae as strong gravitational
regimes whereas the Solar System as weak field regime. We argue that the
inclusion of the quadrupole parameter leads to the modification of the
well-known results coming from the spherical solution due to the Schwarschild
space-time. Hence, we show that in the Solar System regime, considering the
Earth and Sun, there is a weak probability to detect deviations from the flat
case, differently from the case of neutron stars and white dwarfs in which this
probability is larger. Thus, we heuristically discuss some implications on
constraining the free parameters of the phase shift by means of astrophysical
neutrinos. A few consequences in cosmology and possible applications for future
space experiments are also discussed throughout the text.
