No, not any more. The name Area 51 originated from old maps of the Nevada Test Site from the 1950s and 1960s. On these maps, the land around Groom Lake was marked as Area 51.
When it was built in 1955, Kelly Johnson of Lockheed called the Groom Lake base Paradise Ranch in order to attract workers to the U-2 project. Someone at the Central Intelligence Agency, however, gave it the name Watertown, after CIA Director Allen Dulles' birthplace: Watertown, New York. In June of 1958, in preparation for the CIA's OXCART project (the Lockheed A-12), a 38,400 acre section of land around Watertown Airstrip and Groom Lake was legally withdrawn from public use and designated Area 51. This designation was officially used until the late 1970s, during the time when the base was mostly controlled by the CIA. After the Air Force took over in the late 1970s, the name changed to Detachment 3, Air Force Flight Test Center (or simply Det. 3,AFFTC), a remote operating location of the Air Force Flight Test Center at Edwards Air Force Base, CA. Today the official name of the Groom Lake base appears to be the National Classified Test Facility. The Test Wing (Det. 3, AFFTC) is the primary organization and is responsible for operating the facility.
However, the general public commonly uses the term Area 51 to refer to the base. Other names include Dreamland, the Ranch, Red Square, the Box and several others.