The birthdate of 10/11/1784 is given in The Heraldry of the Murrays below - the same publication has differing information for the birthdate of brother George Bishop of Rochester

Burkes Peerage Extract (Full document attached)

2a George (Rt Rev); (105d) born 30 Jan 1761; Bp St David's; married 18 Dec 1780 Anne Charlotte (died 27 April 1844), Lady-in-Waiting to QUEEN CHARLOTTE, daughter of Lt-Gen Francis Grant, MP (see STRATHSPEY, B), and died 1803, having had, with other issue:

• 1b John; RN; died 1803

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http://www.archive.org/stream/heraldryofmurray00john#page/22/mode/2up Page 23 No 105(d)1

105(d) George Murray, born 30th January 1761; died 3rd June 1803. In 1801 he was made Bishop of St. David's. He married, 18th December 1780, Anne, daughter of Lieutenant-General Francis Grant, and had:

• 105(d)1 Captain John Murray, R.N., born 10th November 1784 ; died December 1803.