This happens to be tangentially related to some research I'm currently doing. How exactly George Washington himself knew about Tibet, that I specifically can't answer. What I can say is that by the 18th century, Tibetan Buddhism wasn't exactly completely unknown to Europe.
In the beginning of the 17th century the western Mongolian Oirats began a steady migration to the lower Volga region, coming in contact with the Imperial Russian state. They would come to be known as the Kalmyks, and formed a Khanate from 1630–1771, when Catherine the Great incorporated the Kalmyk holdings into the Russian empire. The Kalmyks moved westwards towards Russia for a variety of reasons, confined territories amongst rising feudal competition, sparse population density in the west Urals and Volga region, favorable trading opportunities with the Russians, and some diplomatic buffers to offset Nogai and Crimean Tatar power proved incentivizing for the Russians. With time, the Kalmyks swore oaths of fealty and arranged treaties with the Russians, this was not a mere conquest by the Russian Empire, but a gradual process of voluntary incorporation. Kalmyks contributed to Russian military endeavors in the 18th century onwards, such as the Northern War and Russo-Turkish Wars (eventually too some Kalmyks even joined the Coalition Powers against Napoleon.) Whatever the case, the Kalmyks, through the Russian Empire, became a greater part of the international stage with its proximity towards Europe. Indeed, Russia had long contact with Mongolian powers historically, but the Kalmyks proved a closer and more concrete relationship. Moreover having read some of Catherine's memoirs I recall she mentions a number of Kalmyk servants and maids for the aristocracy.
The relevance of the Kalmyks and Mongolian powers in this time is their close adherence to Tibetan Buddhism. Mongolians had began adopting Tibetan esoteric and tantric Buddhism during the 13th century, as the religion offered political and religious legitimization for the Khans, as well as power balances against established systems such as in China. Oirats in particular followed the Gelug school of Tibetan Buddhism, a major branch which the Dalai Lama is the spiritual head. Under the Khoshut Khanate Tibet was unified under the authority of the 5th Dalai Lama. A later Oirat power, the Dzungars, toppled the Khoshuts but continued political relations with the Dalai Lama. The Kalmyks were no exception, one of their greatest leaders Ayuka Khan had been confirmed his title by the Dalai Lama, and it was Ayuka Khan that Peter the Great had personally met to enlist Kalmyk support against the Persians.
There's much more than can be said about the Kalmyks, Oirats, and Tibetans, but the important thing here is that as devoted Buddhists, the Kalmyks and other Mongolian groups in the Russian Empire offered early European contact with Tibetan Buddhism. Given the obvious importance of the United States forming diplomat relations with international powers such as the Russians, it's not unlikely for Tibetan Buddhism to be mentioned.
Also, not to fault the first President's lack of knowledge on Tibetan affairs, but it's somewhat inaccurate to think of "eastern Lamas" as shut off from society. Certainly many monastics were secluded in remote areas, but as with most Buddhist societies, temples formed a deeply close relationship with the lay community. Lamas were a part of everyday life, they offered spiritual guidance and religious duties, in exchange for the monetary and material support of the community they serviced.
The Dalai Lamas themselves were rather important political leaders. This thread contains a very educational and worthwhile read on Tibetan society from u/JimeDorje.
Recommended readings on the Kalmyks
This quotation originates not with George Washington himself, but rather in the journal of William Maclay, a Senator for Pennsylvania, in the entry for 4 May 1789:
I told the General [Arthur St. Clair] that General Washington stood on as difficult ground as he ever had done in his life: that to suffer himself to be run down, on the one hand, by a crowd of visitants so as to engross his time, would never do, as it would render the doing of business impracticable; but, on the other hand, for him to be seen only in public on stated times, like an Eastern Lama, would be equally offensive.
There are any number of ways Maclay might have encountered lamas in his readings, none of which he specifies. Nor is it entirely clear if he even necessarily understood what lamas were at any length. However, there are certainly quite plausible ways he might have heard of them. To make a bit of a disclaimer before we start, I found most of the examples below through ECCO, a service that specifically digitises British books of the eighteenth century. However, as these serve not as a precise chronology but rather as an illustrative sampling, and because there would have been some transmission of books across the Atlantic, this should not be too problematic for the purposes of this answer.
What seems to me the most likely explanation for why lamas were fresh in the mind for Maclay its that it was thanks to a visit by the British diplomat Samuel Turner to Tibet in 1783-4, where he met the then 18-month-old 7th incarnation of the Panchen Lama (identified as the Téshoo Lama). Turner’s full account would not be published until 1800, but the journey was at least reasonably known to British audiences by the end of the decade: the playwright Richard Cumberland, in the second volume of his collected essays published in 1786-90, included an essay ‘On the lama of Tibet’ in which he claimed that the Panchen Lama visited by Turner was checks notes the reincarnation of Noah and that one of their intermediate incarnations had witnessed Pythagoras at work and would be able to reproduce his lost work. But questionable theology aside, it does show that the Anglophone world did have a recent episode of direct exposure to the Tibetan world, and to one of its most senior lamas, and that this episode was at least somewhat known among public figures.
However, a certain familiarity with Tibetan Buddhism, at least in name, had existed well before the return of Turner’s mission to British India. It is often forgotten that Gibbon’s History of the Decline and Fall of the Roman Empire continues into the demise of the Eastern Roman Empire as well as the Western, and as part of that he makes allusion to the Tibetan lamas in his brief narrative of the rise of Chinggis Khan, and also to the policy of his grandson Khubilai during his rule of China:
Under the reign of Cublai, letters and commerce, peace and justice, were restored; the great canal, of five hundred miles, was opened from Nankin to the capital: he fixed his residence at Pekin; and displayed in his court the magnificence of the greatest monarch of Asia. Yet this learned prince declined from the pure and simple religion of his great ancestor: he sacrificed to the idol Fo; and his blind attachment to the lamas of Thibet and the bonzes of China provoked the censure of the disciples of Confucius.
Gibbon’s footnote, however, is revealing as to the limitations of this knowledge:
The attachment of the khans, and the hatred of the mandarins, to the bonzes and lamas (Duhalde, Hist. de la Chine, tom. i. p. 502, 503) seems to represent them as the priests of the same god, of the Indian Fo, whose worship prevails among the sects of Hindostan Siam, Thibet, China, and Japan. But this mysterious subject is still lost in a cloud, which the researchers of our Asiatic Society may gradually dispel.
'Fo' is quite plausibly 佛 fo, which is simply the term for ‘Buddha’, which gets across just how poor the Western awareness of the actual theology of Tibetan Buddhism was at this time. But there are two critical things here: the first is that Gibbon did at least vaguely grasp that lamas were clerics. The second is that while the final volume of Gibbon’s Decline and Fall was published in 1789, his source, Jean-Baptiste du Halde’s Description de la Chine, was published in 1735 and translated into English in 1737. This long predates Maclay’s comments on Washington, and shows that an at least superficial knowledge of the existence and nature of Tibetan clerical institutions had existed in the Anglophone world for half a century by that point. References to lamas later appear, for instance, in Oliver Goldsmith’s satirical work The Citizen of the World (1761), in which he writes a series of letters in the persona of a Chinese philosopher residing in London, and in one he writes as follows:
When a man has once secured a circle of admirers, he may be as ridiculous here as he thinks proper; and it all passes for elevation of sentiment, or learned absence. If he transgresses the common forms of breeding, mistakes even a tea-pot for a tobacco-box, it is said, that his thoughts are fixed on more important objects: to speak and act like the rest of mankind is to be no greater than they. There is something of oddity in the very idea of greatness; for we are seldom astonished at a thing very much resembling ourselves.
When the Tartars make a Lama, their first care is to place him in a dark corner of the temple; here he is to sit-half concealed from view, to regulate the motion of his hands, lips, and eyes; but, above all, he is enjoined gravity and silence. This, however, is but the prelude to his apotheosis: a set of emissaries are dispatched among the people to cry up his piety, gravity, and love of raw flesh; the people take them at their word, approach the Lama, now become an idol, with the most humble prostration; he receives their addresses without motion, commences a god, and is ever after fed by his priests with the spoon of immortality. The same receipt in this country serves to make a great man. The idol only keeps close, sends out his little emis|saries to be hearty in his praise; and straight, whether statesman or author, he is set down in the list of fame, continuing to be praised while it is fashionable to praise, or while he prudently keeps his minuteness concealed from the public.
While this is far from likely to be the source of Maclay’s impression of lamas as concealed except for ceremonial needs, it is nevertheless illustrative of it, and suggests that said impression was a relatively broad one.
Adam Ferguson’s 1767 ‘Essay on the History of Civil Society’ invokes the lamas as a form of primitive clerical headship congruous with the druids of Iron Age Europe, but with the difference being that the druids were sources of rudimentary civil government while the Lamas began the introduction of despotism – what distinguishes these, he does not make clear in the immediate context:
In this wild and lawless state, where the effects of true religion would have been so desireable [sic], and so salutary, superstition frequently disputes the ascendant even with the admiration of valour; and an order of men, like the Druids among the ancient Gauls and Britons, or some pretender to divination, as at the Cape of Good Hope, finds, in the credit which is paid to his sorcery, a way to the possession of power: his magic wand comes in competition with the sword itself; and, in the manner of the Druids, gives the first rudiments of civil government to some, or, like the supposed descendent of the sun among the Natchez, and the Lama among the Tartars, to others, an early taste of despotism and absolute slavery.
But this, along with Goldsmith's work above, both serve to illustrate what the sort of 'stereotypical' view of lamas would have been among British intellectual sorts by the 1760s.
Something worth noting here is that information about lamas was very much filtered through China, but was complicated by the understanding that they were not a Chinese phenomenon. The Qing Empire, which ruled over both China and most of the Vajrayana Buddhist world – i.e. Tibet and Mongolia – was uniquely positioned to enable the spread of information about Tibet and its religious institutions to maritime European powers with access to Chinese ports. As you may have noticed, lamas are frequently referred to as the priesthood of the ‘Tartars’ – a broad-brush term for Inner Asians – and thus uniquely barbaric and Other, whereas China and the Chinese tended to be viewed as a more comparable fellow civilisation for most of the eighteenth century.
So while Maclay may have known very little actual information about Tibet or its priesthood, he would almost certainly have had exposure to at least a common set of cultural tropes around lamas: notions of their seclusion, the ceremonial surrounding them, and their specifically being part of a barbarous ‘Tartar’ world diametrically opposite to that of ‘civilised’ Europe and its settler-colonies, and thus ideal to invoke as a negative exemplar. Whether he had any awareness of Samuel Turner's mission, or whether it had any influence on his view, is another matter entirely, and one that the simple brevity of Maclay's quote leaves basically impossible to really discuss. But I would suggest that it is certainly plausible that it may have influenced why he even chose to make the allusion he did.