Born in Newcastle upon Tyne in the North of England, Graham had left Tyneside to find work in Newport. While in Newport he joined the rugby club, making his debut on 15 October 1886 against the Cardiff Harlequins. In 1888, Graham was part of the Newport team to face the first Southern hemisphere touring team the New Zealand Māoris. Led by Theo Harding, Newport were out-muscled by the Māoris, losing three tries to nil. By 1890, Graham had been made the team captain, an honour he would hold for four consecutive seasons.