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Usually preceded by the definite article the, the expression Quatorze Juillet is a borrowing from French, designating the fourteenth of July, the national holiday of the French republic, commemorating the storming of the Bastille on 14^th July 1789 and the Fête de la Fédération, which was held on 14^th July 1790 to celebrate the French Revolution and the unity of the French people. The fourteenth of July became an official holiday in 1880.

—Synonym of Quatorze Juillet: Bastille Day.

The expression Quatorze Juillet has been shortened to Quatorze, usually preceded by the definite article the. The following, for example, is from Club News, published in the Stanmore Observer (London, England) of Thursday 2^nd August 1990 [page 19, column 1]:

CHANTECLER—FRENCH CIRCLE of HARROW: The Circle recently held the annual petanque championship. Roger and Simone Guibaud organised the event and provided some welcome hospitality.
[…]
Because it was “Le quatorze”, the afternoon ended with the singing of La Marseillaise.

These are, in chronological order, the earliest occurrences of the expression Quatorze Juillet that I have found:

1-: From the San Francisco Chronicle (San Francisco, California, USA) of Monday 3^rd July 1882 [page 1, column 2]:

THE FRENCH FETE.
Arrangements for Celebrating the Fall of the Bastille.

What the Fourth of July is to the American citizen the fourteenth of the same month is, in a great measure, to the French resident. The first commemorates the signing of a document which marked the commencement of a republican nation; the second is the anniversary of the destruction of a prison that was the embodiment in stone and iron of monarchial injustice and oppression. The fall of the Bastile [sic] has latterly been looked upon as the fitting point from which to date a new era, and each year the event is celebrated by the “people” of France with becoming ceremony and rejoicing. Whether at home or abroad the representatives of the French republic maintain the observance of the ”Quatorze Juillet,” and not to be behindhand in this now national celebration, the Gallic portion of San Francisco’s population have decided to keep up the day here and show their brothers in Europe that, though in a foreign land of adoption, they do not forget their home.

2-: From The Daily Telegraph (London, England) of Saturday 15^th July 1882 [page 3, column 1]:

FRENCH NATIONAL FETE.
[TELEGRAM FROM OUR CORRESPONDENT.]

PARIS, Friday Night.
Paris is celebrating to-day her third national fête with a zeal and spirit worthy of her old reputation. Every one is out and about engaging in the many divertisements provided by the Municipal Committees, and certainly there is no lack of amusement for persons of all ages and tastes. Lovers of operatic and theatrical performances have had a long list of representations, all free, from which to make a selection, and the crowds which have formed at the doors hospitably opened in honour of the Quatorze Juillet have testified in an unmistakable manner how much this privilege is appreciated.

3-: From The Los Angeles Times (Los Angeles, California, USA) of Sunday 14^th July 1889 [page 4, column 2]:

THE CENTENNIAL OF FRENCH LIBERTY.

Today is the hundredth anniversary of the first great break for liberty made by the French as a nation.
Since the inauguration of the French Republic, the quatorze Juillet has taken the place, as the national day of France, of the quinze Aout—the 15^th of August—“Napoleon’s day,” which, during the third empire, was celebrated throughout France with such pomp and circumstance.

The earliest occurrence that I have found of Quatorze, shortened form of Quatorze Juillet, is from Occasional Notes, published in The Pall Mall Gazette (London, England) of Wednesday 13^th July 1898 [page 2, column 3]:

To-morrow they will be celebrating the National Fête again, over the way. This cult of the Quatorze had become curious years ago, Gambetta applied one of his energetic phrases to the function—“haillon de guerre civile,” ¹ he called it; and, of course, there must be any number of Frenchmen who know that their country had a history before the fourteenth of July, 1789. To such the anniversary of the taking of the Bastille probably no longer appeals very strongly. They know all about that famous stronghold of tyranny. They do not enthuse over the rescue of the victims of oppression therein entombed—the four forgers, the brace of lunatics, and the crapulous nobleman. The massacre of its garrison of invalids is no longer a glorious and immortal memory. In fact, Frenchmen as a whole perfectly understand that the Bastille business was an anarchical proceeding such as would be promptly and severely nipped in its inception by a Republican Minister of the Interior of to-day. They would readily drop the Quatorze if they knew how, or if they had anything to put in its place. Meantime, they are experimenting to-day with a post-dated Michelet ² celebration.

¹ In fact, it was not about the French Revolution (1789-99) that the French stateman Léon Gambetta (1838-1882) used ce haillon de guerre civile (i.e., this rag of civil war), but about the Commune, the revolutionary government established in Paris from 18^th March to 28^th May 1871. Gambetta used ce haillon de guerre civile in a speech delivered on 21^st June 1880 in the Chamber of Deputies, advocating an amnesty to the Communards.
² Jules Michelet (1798-1874) was a prominent French historian and author, best known for his multivolume Histoire de France.