He was the eldest son of Edward Conyers, MP and was educated at University College, Oxford (1735). He succeeded his father in 1742, inheriting a somewhat dilapidated Copt Hall, near Epping, Essex, which he demolished and rebuilt.
He married twice; firstly Hannah, the daughter of Richard Warner, of North Elmham, Norfolk and secondly his cousin, Lady Henrietta Frances, the daughter of Thomas Fermor, 1st Earl of Pomfret. They had 13 children, of whom 8 survived.