Especially in two-martini lunch and three-martini lunch, the U.S. phrase martini lunch (also martini luncheon) denotes a midday meal, with several martinis taken as aperitifs, enjoyed by businessmen, and/or politicians, and/or federal-government employees—here, martini designates a cocktail made from gin and dry vermouth.

The first two occurrences of martini lunch(eon) that I have found are from Robert C. Ruark’s ¹ column:

1-: Published in several U.S. newspapers in December 1950—for example in The Greensboro Record (Greensboro, North Carolina) of Tuesday the 12^th:

New York, Dec. 12.—George McCadden, an American newspaperman who works in Australia, and who has not been home since before the war quit, just checked in like a man from Mars, and is not too impressed by the United States. […]
[…]
After having been subjected to the three-hour New York martini lunch, George is almost beginning to yearn again for the peace of the Australian pub, where the barmaids have character and a limited time is allowed for the clients to get shikkered ².

¹ Robert C. Ruark (1915-1965) was a U.S. columnist.
² The Australian-English adjective shikkered, also shickered, means drunk, intoxicated.

2-: Published in several U.S. newspapers in September 1951—for example in the Citizen-News (Hollywood, California) of Thursday the 27^th:

New York, Sept. 27.—This rickety old frame is racked with laughter over the story of the expulsion of a testifier in the RFC fraud-possibility hearings. The Senate Investigating Committee, at the suggestion of Sen. Joe McCarthy ³ gave the boot to Cecil Green, Lithofold Corp.’s Washington man, because “he had a martini for lunch and was in no condition to testify.”
I don’t know whether the man was loaded, but I sure would have hated to take a breath check on the peers who were present at his dismissal. That Washington town, if anything, is a more fanatic devotee of the three-hour martini luncheon than New York, which fights the midday booze pretty good.
[…]
I do not fancy the martini luncheon myself, being a person of considerable purity, but it has come to be as much a part of the American big-wheel scene as the expense account and the softening-up celebrity party. The flossier traps around the nation’s major cities are jammed, at lunch, with people who are all trying to sell each other something, and are using juniper juice as a lever to the deal. This results in a national drowsiness in the afternoon, but is widely practiced, nevertheless.

³ From 1950 to 1954, the U.S. Republican senator Joseph Raymond McCarthy (1909-1957) led widespread investigations into alleged communist infiltration in U.S. public life.

Rather curiously, in the following from Resolutions I wish they’d make, published in The American Weekly (New York: Hearst Publishing Company) of Sunday 28^th December 1952, the U.S. columnist Dorothy Kilgallen (1913-1965) associated “a third martini before lunch” with Joe Blow, denoting an average, ordinary man:

As the whistles blow on New Year’s Eve and the calendar presents us battered humans with a clean new page, it is customary for confetti-spattered revelers all over the world to halt the whoopee briefly and make some noble resolutions.
Joe Blow vows that in 1953, by golly, nobody is going to talk him into a third martini before lunch, and Mamie Blow, his devoted frau, swears that this year she will lose 11 pounds or starve in the attempt. Amateur Scrooges give tender thoughts to orphans and panhandlers. And delinquent juveniles promise to count 10 before throwing that rock through the schoolhouse window.

When the U.S. general and Republican statesman Dwight David Eisenhower (1890-1969) took office as the 34^th President of the U.S.A. in January 1953, his administration officially prevented the federal-government employees from having long lunch-breaks. This was first mentioned in the column Washington Calling, published in several U.S. newspapers in January 1953—for example in The Pittsburgh Press (Pittsburgh, Pennsylvania) of Sunday the 4^th:

Memo to staff at Eisenhower Headquarters in New York lays down the law: No more two-hour lunches, leaving headquarters unable to do business. And—only one martini at lunch.

Likewise, Hal Boyle, of the Associated Press, used the phrase three-martini lunch in Coffee Break Threatened: Panic On The Potomac; Will Ike’s Boys Kill The Two-Hour Lunch?, published in The News-Palladium (Benton Harbor, Michigan) of Wednesday 28^th January 1953:

The government goal to abolish the two-hour lunch, of course, will have no effect on the business world. The problem of the business world is to cut down on the three-martini, four-hour lunch.

Also in connection with Eisenhower’s new administration, the phrase two-martini lunch occurs in a United-Press story published in several U.S. newspapers on Monday 2^nd February 1953—for example in The Knoxville Journal (Knoxville, Tennessee):

Washington, Feb. 1 (UP)—The first fortnight of the Eisenhower administration has almost brought about the downfall of an old and cherished capital institution—the two-martini lunch.
This pleasant noontime custom has been banished outright in three government agencies and is disappearing by tacit understanding from others.
New Republican bureau heads have let it be known that the lunch hour for federal employes should consist only of lunch for an hour. Anyone who takes an extra 10 minutes or so to toss off a couple of pre-lunch cocktails is in danger of losing his job.

This led the Los Angeles Times (Los Angeles, California), on Thursday 5^th February 1953, to raise the question of who paid for the federal-government employees’ martini lunches:

Who Got the Tab?
News that the two-Martini lunch has been banned from several government departments and is disappearing in others bursts upon a nation that was not aware its salaried employees were taking that kind of lunch.
Since two Martinis, in a classy bar, would be at least $1.50 and not much of a lunch could be had for less than another $1.50, the question may be asked, who picked up the tab? The average government worker would hardly be able to do it very often.
The new regime, it is said, has emphasized that lunch hour is one hour long and no longer, and those who take an extra 30 minutes or more are in danger. Getting to work late and leaving early, another Washington custom, is also on the way out, it’s said.
The money so saved the taxpayers won’t balance the budget but it will help.

And, in a letter published in the Los Angeles Times (Los Angeles, California) of Tuesday 10^th February 1953, one Mrs. D. E. wrote:

Most government workers are motivated by a deep loyalty to country and stay at jobs where the gain is small because they get a sense of satisfaction from serving.
And may I also say, the invitation to a two-Martini luncheon usually comes from the business representative in town with a few axes he hopes to sharpen.

This Grin and Bear It cartoon by George Lichty (George Maurice Lichtenstein – 1905-1983) was published in The Jersey Journal and Jersey Observer (Jersey City, New Jersey) of Tuesday 17^th February 1953—it depicts a group of federal-government employees reading a notice affixed to a wall, addressed to “all govt employees”, and exclaiming:

“. . . ‘No loafing!’ . . . ‘shorter lunch hours!’ . . . ‘an honest day’s work!’ . . . how do they expect to keep good men in government service that way?”