Graham was the son of Fergus Graham, or Grahme, of Plump, Kirkandrews-upon-Esk, Cumberland, and his wife Sibill Bell, daughter of William Bell, of Scotsbrig, Middlebie, in Scotland. He was Gentleman of the Horse to James I.
In 1624 Graham bought the manor of Norton Conyers in the North Riding of Yorkshire. In 1626, he was elected Member of Parliament for Carlisle. He was re-elected MP for Carlisle in 1628 and sat until 1629 when King Charles decided to rule without parliament for eleven years. He was created baronet of Esk, Cumberland on 29 March 1629.
Graham was a faithful supporter of the Royal cause in the Civil War. He was severely wounded at the Battle of Marston Moor on 2 July 1644. He purchased the estate of Netherby and the Barony of Liddell in Cumberland.
Graham married in or before 1624, Catharine Musgrave, daughter of Thomas Musgrave, of Cumcatch, Cumberland and his wife Susanna. She died on 23 March 1649 and was buried at Wath, at the age of 47.
- William Page, ed. (1914). "Parishes: Wath". Victoria County History. A History of the County of York North Riding: Volume 1. pp. 390–396. Retrieved 11 July 2017.
- George Edward Cokayne Complete Baronetage Volume 2 1900