Francis George Cathcart Graham was educated at the South African College, Cape Town, during 1869-1870. He entered the civil service of the Cape of Good Hope in 1870 as a temporary clerk in the customs department. Other clerical positions followed and he was stationed in Albany from 1874 to 1881, Robertson in 1882, and Paarl from 1883 to 1885. In June of the latter year he was appointed master of the high court of Griqualand West. From June to December 1888 he was acting magistrate and member of the special court at Kimberley, and thereafter officer to visit convict outstations in the region. From April 1890 he was civil commissioner and resident magistrate of the district of Wodehouse in the Eastern Cape, with his seat at Dordrecht. Subsequently he served in the same capacity in Somerset East (1897-1900) and Grahamstown (1900-1910). In 1899 he was appointed as a special commissioner to inquire into agricultural distress in the frontier districts, and during 1901 and 1902 was president of the special high treason commission.
Graham's activities in natural history commenced before or in 1891, when he donated butterflies from Dordrecht to the South African Museum in Cape Town, accompanied by careful notes on the dates and localities of collection. Further donations and notes followed in 1892-1894. In 1897 and 1899 he presented more butterflies, this time from the Somerset East district. And in 1913 he donated butterflies from around Grahamstown to Albany Museum.
By 1906 he had extended his interest to small mammals and during 1907 to 1909 presented Albany Museum with a large series of introduced and native rats, mole rats and other small mammals. His third area of interest was conchology, for in 1911 he presented a very large collection of marine shells from Jeffreys Bay to Albany Museum. Some of his collecting in 1913 was done with "Master Reginald Graham", presumably his son.
Graham joined the South African Philosophical Society in 1897 and remained a member to 1906. In 1908 he became a member of the South African Ornithologists' Union.