Quain is the eldest child of John Quain of Carraig Dhúin (Carrigoon), Cork and Mary, daughter of Michael Burke of Mallow, Cork. He is sent to the Diocesan School at Cloyne for his early education and then, at age 15, apprentices to the surgeon-apothecary Fraser in Limerick for five years. In 1837 he enrolls in medicine at the University College London, where his cousins, Jones Quain (1796–1865), the anatomist, and Richard Quain, FRCS, hold teaching posts. He graduates M.B. with honours in 1840.
In 1842, Quain receives the gold medal for achievements in physiology and comparative anatomy, and later he becomes successively house surgeon and house physician at the University College Hospital and commences practice in London, being in particular a protégé of professor Charles James Blasius Williams (1805–1889). He soon has a busy practice, numbering an important clientele, with contacts to the most highly recognised persons.
Quain is chosen in 1846 to be an assistant-physician to the Brompton Hospital for Diseases of the Chest. He retains his connection with that institution until his death, first as full physician (1855), and subsequently as consulting physician (1875). He holds the same rank at the Seamen’s Hospital, Greenwich, and the Royal Hospital for Consumption in Ventnor.
Also in 1846, Quain becomes a member of the Royal College of Physicians and a fellow in 1851. In 1850, he vacates the Chair of Anatomy at the University College London and is succeeded by George Viner Ellis. He is an early member of the Pathological Society of London in 1862, being elected its president in 1869. He is also a fellow and vice-president of the Royal Medical and Chirurgical Society and the Medical Society of London, as well as President of the Harveian Society of London (1853) and fellow of the Royal Statistical Society. He is elected a Fellow of the Royal Society in 1871.
In 1881, Quain is asked by Queen Victoria to attend prime minister Benjamin Disraeli during his last few days. In 1890, he becomes physician-extraordinary to the Queen, and is created a baronet of Harley Street in the County of London and of Carrigoon in Mallow in the County of Cork, in the following year.
Quain is the author of several memoirs, dealing for the most part with disorders of the heart, but his name will be best remembered by the Dictionary of Medicine, the preparation of which occupies him from 1875 to 1882 (2nd edition, 1894; 3rd, 1902).
Sir Richard Quain dies at the age of 81 on March 13, 1898.