A Gulfstream 450 airplane encountered turbulence while in level cruise flight at 36,000 ft. and a cabin server (CS) fractured her arm and leg. The flight crew reported that after departure, while climbing through 20,000 ft, they encountered unexpected light turbulence, and the seatbelt sign was illuminated. About 30 minutes into the flight at a cruising altitude of 36,000 ft (FL360) in visual meteorological conditions with the autopilot engaged, the copilot noticed a cloud top “rising up” from under the nose of the airplane. The airplane “pitched up and down rapidly with a hard jolt;” however, the autopilot did not disengage, and the airplane returned to normal flight. A passenger notified the flight crew of injuries to the CS. At the time of the turbulence event, the CS was in the aft galley taking catering notes, when she felt a rising sensation. She attempted to get low to the floor in case turbulence was encountered. While looking over her shoulder to check on the passenger cabin, she was thrown into the air and “hurled” to the floor impacting on her left shoulder, arm, and hip. Upon being notified of the injury, the flight crew declared an emergency and diverted to Miami for medical assistance. The injured CS was transported to the hospital where she was diagnosed with a fractured left arm and multiple fractures to her left leg. At the time of the turbulence encounter there were no active convective significant meteorological information (SIGMETs), international SIGMETs, center weather advisories (CWAs) or pilot reports (PIREPs) for the area where the turbulence occurred.