We theoretically study the dynamic dielectric response function of a gas of massless Fermions embedded in a coupled double quantum wire structure based on graphene. We write the dielectric function within the random phase approximation (RPA). We approach the system using the two-dimensional (2D) Dirac-like Hamiltonian in the first place, where a parameter β, accounting for the interaction between the substrate and the graphene sheet, is considered in an ad-hoc manner. We study the weak tunneling regime between the two ribbons and find the energy dispersion of the acoustical and optical plasmon modes. Our results show that different choices for the parameter β in the structure should induce spatial anisotropy effects on the plasmon modes.