I have actually written a response on this subject before! Reproduced below:

Possibly! It's not that barbershops were a hotbed of close, four-part harmony - but there was a tradition of African-American "barber musicians" in the nineteenth century. With rampant segregation preventing black musical groups and singers from using most other community buildings, barbershops became social centers for various activities, including rehearsal. Lynn Abbott quotes from a number of newspaper articles that give information about men who owned tonsorial parlors and ran orchestras or gave lessons, in Out of Sight: The Rise of African American Popular Music, 1889-1895:

DECEMBER 20, 1890 (CHICAGO, ILLINOIS): “Prof S. W., East 21st street, proprietor of the ‘Little Jem Palace’ is not only a knight of the razor, but controls the oldest and largest colored orchestra in Chicago. Any number of pieces can be secured on short notice for entertainment, receptions, etc. Address above number” (Indianapolis Freeman)

JANUARY 15, 1892 (ADRIAN, MICHIGAN): At No. 5, East Maumee street is the well known barbershop of Messrs. Jackson and Reid. Their parlors are fitted up with handsome chairs and other accompaniments to a first class business, and it is one of the handsomest shops in the city. They are skilled workmen and enjoy a large and deserved trade. At the rear of these parlors is a small room which has been fitted up with glass cases which are filled with a fine collection of musical instruments. Mr. B. F. Jackson, the proprietor, gives instruction upon 10 or 12 instruments, and has at present 25 or 30 pupils. He and his three children furnish music for many of the best entertainments given in the city” (Detroit Plaindealer).

JANUARY 21, 1893 (DECATUR, ILLINOIS): “Plays in a White Orchestra.” This article describes one of “Decatur’s men of integrity,” J. W. Anderson, “the popular tonsorial artist. He came to Decatur less than three years ago, opening a barber shop and hair dressing parlor, where he has continued to prosper until his trade, both white and colored, became so large that it was necessary to seek a larger room. He purchased the present site, 319 North Water street, the principal thoroughfare of the city. He is also an expert with stringed instruments, such as banjo, guitar, mandolin and violin. He is also a member of the opera house orchestra (white)” (Indianapolis Freeman).

While none of the articles Abbott came across specifically referenced quartets singing in barbershops, W.C. Handy and James Weldon Johnson claimed in the 1920s to have remembered seeing and hearing them - it's very likely, as popular ballads of the period were often written in two- or four-part harmony, though in a Soprano-Alto-Tenor-Bass configuration where the soprano part had the melody, rather than the Tenor-Lead-Baritone-Bass configuration used in barbershop quartet, with the lead having the melody. (You can see a Victorian music book here.) By the 1890s, however, writers were referring to a "barbershop chord" that implied the sound that we also think of as barbershop: this was "a secondary dominant chord that appears on the flatted sixth degree of the scale," according to Vic Hobson in Creating Jazz Counterpoint: New Orleans, Barbershop Harmony, and the Blues. It doesn't really appear in classical European music, and may have developed from the practice of "cracking up a chord", one person singing a note and the others forming a chord around it, which is itself the traditional way of developing the harmony parts in barbershop. Some other characteristics of this new version of the white quartet singing were improvisation, and embellishments like sliding from one note to another that had been handed down from slave song traditions. The style became very popular for professional black quartets on the stage, as well as minstrel shows, and began to cross race lines.

As everything passes, barbershop singing fell out of style in favor of the blues and jazz, and it stopped being practiced. Recognizing that people would forget how to do it, Owen Cash began the facetiously named Society for the Preservation and Encouragement of Barber Shop Quartet Singing in America in 1938 - it wasn't intended to be a real, national society, just an excuse to sing the old songs, but other chapters quickly popped up. The revival was in high gear by 1940, when a quartet competition was held at the New York World's Fair, and popular groups singing new music were frequently incorporating tight harmonies. However, while the style originated in African-American barbershops, black men were deliberately excluded from the revival. Women were also excluded, and formed the all-white companion society, Sweet Adelines International, in the 1940s; an integrated female society, Harmony, Inc. was formed in 1959. (Information on the revival of barbershop comes from The Barbershop Singer: Inside the Social World of a Musical Hobby, by Robert Stebbins.) The racism here isn't surprising: white nostalgia typically portrays the previous period as an idyllic white space, including non-white figures where it must in a fashion that's non-threatening, whether physically threatening or simply threatening to the idea of white supremacy ... such as being center stage in a musical act that white people wanted to practice. Nostalgic media of this period was happy to depict minstrel shows as good old-time fun, but nothing would star black actors.

(Side note: the "barber garb" is basically leisure wear of the 1910s - the striped coats were popular for summer dress, as were straw hats. It's likely that these came back as the style for the Gay Nineties revival and then became attached to the related musical revival. If you want to read more about the fascination people in the mid-20th century had for the late Victorian and Edwardian periods, I have an answer here.)