Classical and textcolor{blue}{Quantum} Field-Theoretical approach to the non-linear q-Klein-Gordon Equation

\color{blue}{In the wake of efforts made in [EPL {\bf 97}, 41001 (2012)] and [J. Math. Phys. {\bf 54}, 103302 (2913)], we extend them here by developing the conventional Lagrangian treatment of a classical field theory (FT) to the q-Klein-Gordon equation advanced in [Phys. Rev. Lett. {\bf 106}, 140601 (2011)] and [J. Math. Phys. {\bf 54}, 103302 (2913)], and the quantum theory corresponding to $q=\frac {3} {2}$. This makes it possible to generate a putative conjecture regarding black matter. Our theory reduces to the usual FT for $q\rightarrow 1$.}