Born in Hungerford, Berkshire, he joined the Democratic Federation (forerunner of the Social Democratic Federation) in 1881 and was elected to its executive in 1883. When much of the party's hierarchy left to form the Socialist League, he remained a supporter of H. M. Hyndman.
Quelch became the full-time editor of the SDF's newspaper, Justice. He also represented the SDF on bodies including various strike committees and the Trades Union Congress, and at socialist conferences across Europe. He was elected several times as the chair of the London Trades Council, and was one of the founders of the Labour Representation Committee (although he supported the SDF withdrawing in 1901). Also in 1901, Quelch arranged for the SDF to print Lenin's newspaper, which had been banned in Russia.