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I ran into this expression when reading Jon Meacham's Art of Power,

He(Jefferson) was usually a master of his emotions. "I know of no gentleman better qualified to pass over the disagreeables of life than Mr. Jefferson, as he makes his calculations for a certain quality of imposition which must be admitted in his intercourse with the world," said a friend of Jefferson's. "When it shows itself in high colors, he has only to count ten and he is prepared for the subject."

I have checked the word "color" in Merriam-Webster, it contains the meaning of "nature, character" when used as plurals. But I feel it's not quite right and I don't know which definition is better fitting here in this context.

'in high colors' in the 1700s and early 1800s

I looked at a number of instances of "in high colors [or colours]" from the 1700s and early 1800s—encompassing the period when William Smith, John Adams's secretary, made the comment about Jefferson that the poster cites—and in all of them the expression appears to be not idiomatic, but literally or figuratively descriptive. In effect, "in high colors" means "in bright colors" or "vividly."

Here are some early instances of its use.

From a letter to Lord Hardwicke, dated September 5, 1758, in John Nichols, Illustrations of the Literary History of the Eighteenth Century, volume 1 (1817):

I must write a book, and wish you had such a leisure from your military cares—I think you would compound with my Doctor for another Historico-political preface, but seriously from an officious paragraph, setting forth in high colours the alacrity of your provincials, in taking commissions; I half concluded that you had found them rather backward.

From a review of Cornelius Cardew, "A Sermon, Preached at Truro, before a Provincial Grand Lodge of Free and Accepted Masons," in The London Review (October 1779):

In this discourse Mr. Cardew stands forth as a warm advocate for the tenets of Free-Masonry, and he hath drawn the character of that institution in high colours.

From "Monthly Register: House of Commons, Tuesday, Feb. 16," in The Literary Magazine, and British Review (March 1790, page 222):

He [Mr. Burgess] then mentioned several circumstances, painting in high colours the necessity of redressing the grievance of arrests according to the powers vested in attornies and creditors. Among which he related an account of a man who was married to an amiable woman, by whom he had eight children; and having grown tired of her, or wishing to marry or live with another woman, had actually been advised by an attorney, whom he procured for his detested purpose, to arrest her, and throw her into Newgate, with an intention of forcing her to come to an accommodation with him for a paltry separate maintenance; and this man was still in practice in the Courts.

From John M'Donald, Letters Addressed "To the Friends of Religion:" Criminating the Presbytery of Albany, and the Synod of New-York and New-Jersey, with Answers, by Jonas Coe (1801):

The offence [receiving Mr. M'Donald as a minister of the Gospel] is painted in high colours. It is to act contrary to the known and certain rules of the Gospel. This is a heavy charge, and required some proof. But why prove what is intuitively known and certain. They add "it is contrary to all the rules of church government and discipline."

From the entry for Vera Cruz [Mexico] in Antonio de Alcedo & ‎George Alexander Thompson, The Geographical and Historical Dictionary of America and the West, volume 5 (1815):

Millinery goods made in Paris. Superfine French woollen cloths, formerly excelling in blacks, blues, as well as in high colours, such as scarlets, roses, crimsons, &c. Hats, both white and black, manufactured in Paris; they are particularly calculated both for Old Spain and Spanish America.

From Henry Crowe, Zoophilos, or, Considerations on the Moral Treatment of Inferior Animals, second edition (1820):

Indeed, where there is any stock in hand of this article of trade [horses sold for dog's meat], (which is in some places carried on upon a large scale,) part of its value must obviously depend upon the length of time that life can be supported in this deplorable state. These scenes I have known represented in high colours, and even with additional horrors; but I will not run a risque of overcharging the picture, by stating more than well-known facts.

From William Webb, an entry dated July 23, 1823, in Minutes of Remarks on Subjects Picturesque, Moral, and Miscellaneous (1827):

I recur to the palaces, in the near survey of which the cause is seen of that brilliant coloring, which is so striking in the remote view of the Genoese buildings. They are not merely coated with a rich stucco, imitative of marble, but many of them are painted, in high colors, with mimic columns and other decorations of architecture and sculpture.

From Sir George Head, Forest Scenes and Incidents, in the Wilds of North America (1829):

These people [groups of "native Indians"] attracted my earnest attention, for my imagination had painted in high colours the interesting spectacle of man in a state of rugged nature, wild as his native woods, and combining with human intelligence the physical strength of the brute creation. It was not, therefore, without considerable disappointment, that I saw a few squalid, miserable-looking beings, straggling in idle listlessness about the streets, and inferior in point of appearance to the wandering race of gipsies in England.

'in high colors' as used in the quotation about Thomas Jefferson

Merrill Peterson, Thomas Jefferson and the New Nation: A Biography (1970) discusses William Smith's description of Jefferson in the context of Jefferson's role in 1789 as U.S. minister to France:

Jefferson had impressive talents for diplomacy. ... His self-possession was remarkable. William Smith, Adams's secretary, said he knew no man better qualified to pass over difficulties because "he makes his calculations for a certain quantity of imposition, which must be admitted in his intercourse with the world. When it shows itself in high color, he has only to count to ten and he is prepared for the subject. Happy state of mind—." "The quantity of imposition" at Versailles was tolerable. Jefferson genuinely liked his position, preferring it to a domestic one more exposed to the buffetings of politics.

As Merrill Peterson indicates, the thing showing itself in high colors was "imposition"—which in the 1700s could mean, among other things, the state of being intruded upon, or inconvenienced, or compelled to perform or accept some obligation, or lied to. Samuel Johnson, A Dictionary of the English Language (1756) lays out the possibilities in his entry for imposition:

IMPOSITION, s. {imposition, French} 1. The act of laying any thing on another. 2. The act of giving a note of distinction. 3. Injunction of any thing as a law or duty. 4. Constraint ; oppression. 5. Cheat ; fallacy ; imposture.

It seems clear, then, that Smith uses "in high colors" to describe an imposition of a particularly obvious, inescapable, and disagreeable sort that may unexpectedly "show itself"—and to express admiration at Jefferson's ability to react coolly, practically, and without irritation to such inevitable instances in the life of a diplomat.

The subject of the sentence "When it shows itself in high colors.... i.e. "it", refers to intercourse.(as you correctly point out)
The intercourse referred to will be a fluid concept which varies from time to time. The usage of the words "high colors" refers therefore to that intercourse being particularly vehement or intense at that particular time.