We have considered the behavior of the conductivity or the dielectric function of the one-dimensional (1-D) quantum sine-Gordon system (g2 = 4π) for two models of impurities. We have used a Fermi-Bose relation and a self consistent t-matrix approximation. As for the conductivity at zero temperature, it appears when the concentration of impurities becomes larger than the critical concentration of impurities. It shows the linear increase near the critical concentration. As for the dielectric constant, its behavior is model dependent. We have numerically calculated the frequency dependence of the conductivity at zero temperature. The real part shows the second order increase near the critical frequency. The imaginary part shows the linear decrease from the origin.