In a graphene-based Josephson junction, the Andreev reflection can become specular which gives rise to propagating Andreev modes. These propagating Andreev modes are essentially charge neutral and therefore they transfer energy but not electric charge. One main result of this work is that when the Dirac theory of graphene is deformed into a tilted Dirac cone, the breaking of charge conjugation symmetry of the Dirac equation renders the resulting Andreev modes electrically charged. We calculate an otherwise zero charge conductance arising solely from the tilt parameters $veczeta=(zeta_x,zeta_y)$. The distinguishing feature of such a form of charge transport from the charge transport caused by normal electrons is their dependence on the phase difference $phi$ of the two superconductors which can be experimentally extracted by employing a flux bias. Another result concerns the enhancement of Josephson current in a regime where instead of propagating Andreev modes, localized Andreev levels are formed. In this regime, we find enhancement by orders of magnitude of the Josephson current when the tilt parameter is brought closer and closer to $zeta=1$ limit. We elucidate that, the enhancement is due to a combination of two effects: (i) enhancement of number of transmission channels by flattening of the band upon tilting to $zetaapprox 1$, and (ii) a non-trivial dependence on the angle $theta$ of the the tilt vector $veczeta$.
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