It is stated that mercury is a liquid at STP because the high relativistic mass of its valence electrons stabilizes their orbits and thus prevents the electrons from being shared. If so, liquid mercury would seem to have no "free" electrons; how, then, does it conduct electricity?
Keep in mind that "stabilizing" orbitals does not "prevent" electrons from being shared, only that they would be shared to a smaller extent. The very fact that mercury is a liquid and not a gas shows that there is some bonding in mercury, even if it is weaker than the bonding in most metals. So, it is conductive the same way other metals are, just less conductive. You can look up the conductivity of mercury, and you will find it to be lower than most metals.