We have calculated charge and heat transport coefficients of a hot QCD medium under a weak magnetic field using the relativistic Boltzmann equation in the BGK framework. The modified collision term, preserving instantaneous particle number conservation, leads to enhanced electrical, Hall, thermal, and Hall-type thermal conductivities compared to the relaxation time approximation. We also compare weak and strong magnetic field results, finding significant suppression of transport coefficients in the weak field case. Furthermore, the Knudsen number indicates local equilibrium is maintained at finite chemical potential, while the Lorenz number shows a violation of the Wiedemann-Franz law at low temperature and saturation at high temperature. Since the Lorenz number remains above unity, thermal conductivity dominates over electrical conductivity in this regime.