“Call girls” were, originally, the telephone operators (mostly women) who connected calls. “Call girl” now refers to a sex worker who is available on call.
 
The term “call girl” was used in 1916, in a vice report before the Illinois state senate. An article about New York City’s “call girls” appeared in 1936 newspapers. The book Call Girls of New York was published in 1953.
 
   
Wikipedia: Call girl
A call girl or female escort is a sex worker who (unlike a street walker) does not display her profession to the general public; nor does she usually work in an institution like a brothel, although she may be employed by an escort agency. The client must make an appointment, usually by calling a telephone number. Call girls often advertise their services in small ads in magazines and via the Internet, although an intermediary advertiser, such as an escort agency, may be involved in promoting escorts, while, less often, some may be handled by a pimp. Call girls may work either incall, where the client comes to them, or outcall, where they go to the client.
 
(Oxford English Dictionary)
call-girl  n. (orig. U.S.) a prostitute who makes appointments by telephone; also attrib.
1940   in Amer. Speech (1942) 17 204/1   Call Girls Die Young.
1951   E. Kefauver Crime in Amer. (1952) xvii. 193   The hotel..has shown up in some of the call-girl operations.
       
20 January 1916, Evening World-Herald (Omaha, NE), “Poverty Cause of Vice Asserts Illinois Board Urging $8 Wage,” pg. 2, cols. 2-3:
Springfield, Ill., Jan. 19.—Poverty is the principal cause of immortality, a minimum wage for girls and women is $8 a week, and unregulated conditions of domestic employment render the home, in many cases, a breeding place of commercialized vice, according to the Illinois senate white slave investigation committee’s report, made public tonight when formally presented to the state senate.
(...)
A condition which, the report declares, strikes directly at the home, is found in what is termed the “call girls” system. On this subject the report says in part:
Society Cafes Reported.
“A detective told of a ‘call’ list which he had seized in a raid. More than twenty names were on the lists, first names only being given, then opposite the telephone numbers. He checked up the names and numbers. Some of the women were ‘respectable’ married women. Two were young daughters. Others were working girls. The case of a young mother serving as ‘call girl’ and using the money she made in buying necessities for her baby is merely illustrative of the character of some of the women in this system.”
 
Chronicling America
20 January 1916, Ogden (UT) Standard, pg. 8, col. 4:
IMMORALITY IN ILLINOIS HOMES
White Slave Investigation Includes the “Call Girl” and Domestic Servants.
 
9 November 1936, Alton (IL) Evening Telegraph, pg. 1, col. 3:
Promised ‘Call’ Girls Wealthy Mates—Charge
NEW YORK, Nov. 9, (AP)—Federal agents called Lucille Malin, 35, for arraignment today, accusing her of operating a call house that dangled a lure of rich prospective husbands befoe the irls on its lists.
 
Rhea Whitney, in charge of the Federal Bureau of Investigation here, described her as “the largest call house operator in New York.”
 
UNZ.org 
6 February 1937, The Saturday Review, “Rats, Lice, and Mystery” book review by Duncan Aikman, pg. 5, col. 2:
There, he says, “madames” and “call girls” are arrested on sight, and pimps are outlawed.
 
17 March 1937, San Francisco (CA) Chronicle, “Acitivities of the McDonough Brothers,” special sec., pg. 2, col. 5:
The foregoing is only one phase of the prostitution situation. There were in addition, countless apartments where tow or three girls operated independently, as well as numerous “call houses” conducted in apartment buildings and “call girls” working out of such building and hotels.
 
OCLC WorldCat record
Call girl
Author: William McClellan
Publisher: New York, Phoenix Press [1940]
Edition/Format:   Print book : Fiction : English
 
Google Books
Disorganization, Personal and Social
By Ernest Russell Mowrer
Chicago, IL: Lippincott
1942
Pg. 323:
In this transition from the harlot to the geisha, the “call girl” occupies an intermediate position. In fact many a geisha may have started as a “call girl,” built up a clientele upon the basis of her desirability as a companion, and graduated into the upper levels of the geisha class.
 
OCLC WorldCat record
Call Girls of New York.
Author: Roland VANE, pseud.
Publisher: London : Archer Press, [1953]
Edition/Format:   Print book : English
 
OCLC WorldCat record
New York call girl.
Author: Robert Lowry
Publisher: Garden City, N.Y., Doubleday & Co., 1958.
Edition/Format:   Print book : English
 
OCLC WorldCat record
Hos, hookers, call girls, and rent boys : professionals writing on life, love, money, and sex
Author: R J Martin; David Sterry
Publisher: Brooklyn : Soft Skull Press ; [S.l.] : Distributed by Publishers Group West, ©2009.
Edition/Format:   eBook : Essay : English
Database: WorldCat
Summary:
The only thing the writers in this book have in common is that they’ve exchanged sex for money. They’re PhDs and dropouts, soccer moms and jailbirds,
 
Irish Mirror
Legendary Paris ‘Madame’ who supplied prostitutes to JFK, Gaddafi and Brando dies at 92
16:36, 22 DEC 2015 UPDATED 17:29, 22 DEC 2015
BY PETER ALLEN
Fernande Grudet was known as Madame Claude and is credited with coining the term ‘call girl’ thanks to her telephone booking system for VIP clients
(...)
Madame Claude, as Mrs Grudet was known, ended her days in a tiny apartment in Nice, on the French Riviera.
(...)
All used Madame Claude’s innovative telephone booking system, giving rise to the term ‘call girl’.