According to flight attendant (FA) statements, the airplane was climbing to cruise altitude in smooth air. While preparing for meal service, a FA pulled out a beverage cart from its stowed position in the aft galley and it immediately toppled over onto the foot of another FA that was in area. The injured FA was unable to continue her duties and the flight diverted to a closer airport without further incident. The flight attendant was transported to the hospital and diagnosed with an open fracture of her right foot. There were no reported malfunctions or failures with the beverage cart.