Using equilibrium molecular dynamics (EMD), we analyzed the friction between water-methanol liquid mixtures and a non-polar solid wall. Specifically, we calculated the friction coefficient (FC) from the autocorrelation of the shear force exerted on a liquid from a solid wall using two Green-Kubo (GK) equations proposed by Bocquet and Barrat (BB) [Physical Review E, Vol. 49, (1994), 3079], and Huang and Szulfarska (HS) [Physical Review E, Vol. 89 (2014), 032119], and compared these FC values with those obtained in our previous non-equilibrium molecular dynamics (NEMD) work. Both GK equations reproduced the FC dependence on the methanol concentration in the first adsorption layer X_a^MeOH observed with NEMD, but the BB method gave a better estimate of the NEMD results with a difference of 20 % at largest. The independent molecular FCs of water and methanol, which were only extractable with the HS method from EMD simulations, corresponded qualitatively well with the NEMD results regarding the dependency on X_a^MeOH. Though certain discrepancies were observed between the EMD and NEMD results, the EMD had a considerable advantage regarding the computational time required to calculate the FC with a comparable error: the EMD computational time was about 20 times shorter than the NEMD time in the present system.