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The U.S. expression holy laugh designates a laugh by a person in a state of religious fervour. This expression seems to have especially been used in reference to Methodist camp-meetings ¹.

¹ The noun camp-meeting designates a religious meeting held in the open air or in a tent (chiefly among Methodists in America), and usually lasting for some days, during which those who attend encamp on the spot.

The following, for example, about camp-meetings, is from The Dixie Frontier: A Social History of the Southern Frontier from the First Transmontane Beginnings to the Civil War (New York: Capricorn Books, 1964), by the U.S. scholar Everett Newfon Dick (1898-1989) [chapter 18, page 198]:

Sometimes a whole congregation would be thrown into side-splitting convulsions of laughter. When it got started in an audience, everybody would be seized with hearty natural laughter. It would last for hours sometimes. This was known as the “holy laugh.”

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

1-: From Recollections of the Last Ten Years, Passed in Occasional Residences and Journeyings in the Valley of the Mississippi, from Pittsburg and the Missouri to the Gulf of Mexico, and from Florida to the Spanish Frontier; in a Series of Letters to the Rev. James Flint, of Salem, Massachusetts (Boston: Cummings, Hilliard, and Company, 1826), by Timothy Flint (1780-1840) [Letter XXI.—New Madrid: page 239]—the following is about New Madrid and the counties of Cape Girardeau and Sainte Genevieve, in Missouri:

These various races […] are more mixed in their population in this region than elsewhere. As it respects their religious opinions, there are considerable settlements north of Jackson, that came in a body from North Carolina; they are generally Presbyterians, and professors of religion. The Germans, as I have remarked, are generally Lutherans. The Baptists, the Cumberland Presbyterians, and the Methodists have many societies, and the Catholics have a large settlement here, composed of French and Irish […].
One general trait appears to me strongly to characterize this region in a religious point of view. They are anxious to collect a great many people and preachers, and achieve, if the expression may be allowed, a great deal of religion at once, that they may lie by, and be exempt from its rules and duties until the regular recurrence of the period for replenishing the exhausted stock. Hence we witness the melancholy aspect of much appearance and seeming, frequent meetings, spasms, cries, fallings, faintings, and, what I imagine will be a new aspect of religious feeling to most of my readers, the religious laugh. Nothing is more common at these scenes, than to see the more forward people on these occasions indulging in what seemed to me an idiot and spasmodic laugh, and when I asked what it meant, I was told it was the holy laugh! Preposterous as the term may seem to my readers, the phrase “holy laugh” is so familiar to me, as no longer to excite surprise. But in these same regions, and among these same people, morals, genuine tenderness of heart, and capacity to be guided either by reason, persuasion, or the uniform dictates of the gospel, was an affecting desideratum.

2-: From a review of The Author’s Preface to the Improved Edition ²—review published in The Western Monthly Review (Cincinnati, Ohio, USA) of January 1829 [page 477]:

We will […] quote a few of the testimonies against these excesses, which are introduced into this pamphlet. […] Dr. Roberts is very pointed in his testimeny [sic] against the ‘abominable practice of jumping, pointing, dancing, boreing, scratching in the earth and jerking.’ Might he not have added the ‘holy laugh?’

² The reference is to the Improved Edition of Methodist Error; or, Friendly, Christian Advice, to those Methodists, who indulge in extravagant emotions and bodily exercises (Trenton (N.J.): Published by D. & E. Fenton, 1819), by a Wesleyan Methodist.

3-: From A Ramble of Six Thousand Miles through the United States of America (London: Published by Effingham Wilson, 1832), by Simon Ansley Ferrall [chapter 3, page 76]—the following is from the description of a Methodist camp-meeting in Ohio:

A scene now ensued that beggars all description. About twenty women, young and old, were lying in every direction and position, with caps and without caps, screeching, bawling, and kicking in hysterics, and profaning the name of Jesus. The preachers, on their knees amongst them, were with Stentorian voices exhorting them to call louder and louder on the Lord, until he came upon them; whilst their attachées, with turned-up eyes and smiling countenances, were chanting hymns and shaking hands with the multitude. Some would now and then give a hearty laugh, which is an indication of superior grace, and is called “the holy laugh.”

4-: From a letter written by the U.S. educator Henry Barnard (1811-1900), dated Hillsborough, North Carolina, Wednesday 27^th March 1833, published in the Maryland Historical Magazine (Baltimore, Maryland, USA) of December 1918 [page 328]:

My dear Brother—
I wrote you a few lines from Chapel Hill on Monday afternoon. I spent the evening at Prof. Mitchell’s, one of the ablest men of the faculty. […] He told me of an incident which frequently occurs at the camp meetings of the Methodists. The preacher in the midst of a fervent prayer, will all of a sudden burst out into a loud boisterous laugh—as though his soul was rejoiced at the conversion of sinners around him. The most godly of his brethren join with him. This is called the “Holy Laugh.”

5-: From An Essay on the Life of the Right Reverend Theodore Dehon, D.D., Late Bishop of the Protestant Episcopal Church in the Diocese of South-Carolina ³ (Charleston (S.C.): Printed for the author, by A. E. Miller, 1833), by C. E. Gadsden, D.D. [chapter 5, page 153]:

In this diocese, indeed in our country generally, what a wide scope for [Dehon’s] commiseration! The temple, in too many places in ruins, given to the mole and the bat, the uncovered sepulchre, the extensive wilderness, in which, though man has found a habitation, none has yet been found for the Lord, his God; and, may I not add, the crowd gathered on the brink of a river, like worshippers by the Ganges, the frantic motions, the wild scream, the torches flitting in the wood at dead of night, the “holy laugh,” as it is profanely called, the contrast of levity and gloom, of mirth and devotion, of poverty and splendid equipage, of sleepiness and vociferation, the confusion of infant and adult voices, and of singing, praying, preaching, clapping of hands and conversation, in different groups at the same moment; and, finally, the whole scene of what is called a religious meeting, might compel the inquiry of the traveller—can this be a Christian land?

³ This refers to Theodore Dehon (1776-1817).