Of British-English origin, the phrase an officer and a gentleman designates a person embodying the civilised qualities supposedly characteristic of both an officer (i.e., a person in a position of authority in the armed forces) and a gentleman (i.e., a courteous, honourable man).

This phrase occurred, for example, in an homage to the British stage and film actor Alec Guinness (1914-2000), by John Marland, published in The Stage (London, England) of Thursday 10^th August 2000 [No. 6,225, page 6, column 4]:

This self-deprecating and shy actor was notoriously reluctant to acknowledge his immense contribution to the profession (“I gave my best performance, perhaps, during the war—trying to be an officer and a gentleman”).

The texts containing the earliest occurrences of an officer and a gentleman that I have found seem to indicate that this phrase originated in the Articles of War * for the year 1749.
—* The Oxford English Dictionary (online edition, March 2024) defines Articles of War as “a set of regulations drawn up to govern the conduct of a country’s military and naval forces”.

These early occurrences of the phrase an officer and a gentleman are as follows, in chronological order:

1-: From Rules and Articles for the Better Government of His Majesty’s Horse and Foot Guards, and all Other His Forces in Great Britain and Ireland, Dominions beyond the Seas, and Foreign Parts ([s.l.]: [s.n.], 1749) [page 27]:

SECT. XV.
Administration of Justice.
[…]
ARTICLE XXV.
Whatsoever Commissioned Officer shall be convicted before a General Court-martial, of behaving in a scandalous infamous Manner, such as unbecoming the Character of an Officer and a Gentleman, shall be discharged from Our Service.

2-: From The Caledonian Mercury (Edinburgh, Midlothian, Scotland) of Tuesday 19^th February 1751 [No. 4,738, page 2, column 1]:

We hear from Ireland, that the following is a Copy of the Sentence given against Ensign James M‘Donald, of Lord Torrington’s Regiment [now the Earl of Home’s] upon the Proceedings of a general Court-Martial held at Kinsale in Ireland, during Lord Harrington’s Government and Presence there, Nov. 16, 1749, for insulting Captain Douglass of the said Regiment.
‘The Evidence being closed, and the Prisoner being withdrawn, the Court took the Matter before them into Consideration, and do find that Ensign M‘Donald is guilty of an Insult to his commanding Officer (Captain Douglas) by calling him Bashaw at the Head of the Colours;——That the Word Rebel, which was meant to Captain Douglass, is false, malicious, groundless, and injurious; and therefore do find the Prisoner guilty of a Breach of the 25^th Article of War, of the 15^th Section, passed in the Year 1749.’
N B. The 25^th Article of War, of the 15^th Section, is, ‘Whatsoever commissioned Officer shall be convicted before a general Court-Martial of behaving in a scandalous, infamous Manner, such as is unbecoming of an Officer and a Gentleman, shall be dismissed from our Service.’

3-: From The Derby Mercury (Derby, Derbyshire, England) of Monday 13^th March 1752 [Vol. 20, No. 1, page 2, column 1]:

We hear his Majesty has confirmed the Sentence of the General Court-Martial, held on the Trial of a certain Officer, whereby he is declared guilty of behaving in a scandalous Manner, unbecoming an Officer and a Gentleman, by wounding an Ensign of the first Regiment of Foot-Guards, whilst the said Ensign’s Sword was in the Scabbard; and adjudged to be discharged from his Majesty’s Service.

4-: From The Proceedings of a General Court-Martial, held at Guildford, August 9, 1758. On Cornet George Moreland, of the King’s own Regiment of Dragoons, commanded by Major-General the Earl of Albemarle (London: Printed for J. Scott, 1759):

[page 28]: Captain Irvine’s Reply to the Prisoner’s Defence.
Mr. President and Gentlemen of the Court-Martial, I have given in my Charge against Mr. Moreland, for accusing me of the grossest of all Crimes, contrary to the Character of an Officer and a Gentleman, I have proved this Accusation, I hope to the Satisfaction of the whole Court, by Major Bonham’s Evidence, and Mr. Moreland’s own Confession.
[page 40]: Major Bonham being call’d on, deposeth, that Captain’s Irvine’s Character, as an Officer and a Gentleman, was always unexceptionable.

5-: From a review of An Account of the Expedition to the West Indies, against Martinico, Guadelupe, and other the Leward Islands, subject to the French King, 1759. By Richard Gardiner, Esq; Captain of Marines on board his Majesty’s Ship Rippon, on the Expedition—review published in The Critical Review: Or, Annals of Literature (London, England) of November 1759 [Vol. 8, page 391]:

Our author makes honourable mention of every officer of the least distinction, whether in the army or navy, who signalized himself by his courage or conduct on this important expedition. He has done justice to the consummate conduct, resolution, and humanity of general Barrington, the enterprizing genius of brigadier Crumpe, the valour and activity of lieutenant-colonel Melville, of the land-service; and, in the navy, to the youthful ardour of the brave old Leslie, the intrepidity of Shuldam, the impetuosity of Burnet, the courage of Jekyl, and the gallantry of Gayton. In a word, he writes like a scholar, an officer, and a gentleman.