Within the framework of the extended Hubbard model on a hexagonal lattice, electrical conductivity was calculated via Green-Kubo relations from current correlator obtained in a Hybrid Monte Carlo simulation. A region near $V_{00} = 3V_{01}$ is considered and a systematic excess of conductivity over ballistic conductivity known for the model is revealed. Indications of several phases which change each other while $V_{01}$ increases are discovered.