John F.W. Quekett was a son of Professor J.T. Quekett of London and was educated in that city. He emigrated to Natal in 1871 and was elected a Fellow of the Zoological Society of London (FZS) during the early 1880's. By 1888 he was honorary secretary of the Museum and Science Department of the Natal Society, Pietermaritzburg, and three years later was secretary and the first curator of the museum (which was a forerunner of the Natal Museum). Meanwhile he applied for employment in the Natal civil service in 1889, and in its new Forestry Department in 1891.
In June 1895 Quekett was appointed curator of the Durban Natural History Museum, succeeding J.S. Steel*. He also served as secretary and treasurer of the museum's management committee. Owing to ill health he retired in 1909. Though he did some work on Natal moths, his specialisation was in conchology. He went on many field trips with H.C. Burnup* to collect shells and they were the first collectors to obtain shells from the stomach of the Musselcracker fish. These were obtained from a Durban fishmonger, Alex ("Lexy") Anderson. Among others the marine shells Glycymeris queketti and Haliotis queketti, and the non-marine species Gulella queketti and Natalina quekettiana, were named after Quekett.
He became a member of the South African Philosophical Society in 1899 and remained a member when it became the Royal Society of South Africa in 1908. In September 1910 he was elected joint vice-president of the newly established (but short-lived) Natal Scientific Society.