The short answer is: Yes, but they probably would have been Inuit and they would have been treated closer to a prized slave, not what I assume are some generic Cherokee people bowing respectfully to the queen. (Sorry, I don't have a high resolution clip of it, just my vague memory)

The longer answer: Yes...

A brief caveat, most of my research is based in the 1700s and beyond.

Native American people were either brought, or traveled to European countries from the 16th century and beyond for a variety of reasons. At first it was mostly either curiosity on the part of explorers, or as a way to express their dominion over the inhabitants of the new lands upon which they claimed. Much like bringing an exotic animal to show off and gawk at. Many of the first Native American and First Nations people who came to the court of Queen Elizabeth fell into this category.

One of the first occurred in 1576, Martin Frobisher, a privateer (aka pirate) kidnapped an Inuit family on Baffin Island and brought him him to England. He was paraded about and they were seen as an oddity to observe. Their names are disputed, and seem to vary wildly, so it would not surprise me if they were just made up for exhibition purposes. Not much else is known about him as far as I am aware, though they did die shortly after arriving in England. We don't know if they appeared in the Queen's court, but Frobisher was thanked by the Queen shortly thereafter. In the 1600s, English settlers would occasionally kidnap and sell Native American people as slaves as well. Most famously was Tisquantum's (commonly called Squanto), who escaped enslavement in Spain and fled to England, where he became an important figure in early Native American relations. As for what these early people thought of their treatment, I'm not sure, but I can't imagine they cared for being kidnapped, enslaved, paraded about, and so on.

I may be reading into your question about "what they thought of England," but keep in mind that to most Native American people at the time, 99.9% of the United Kingdom would not be terribly impressive to a Native American person. Most English people still lived in small villages filled with wooden and mud homes, farmed, and had little access to weapons or technology that would have seemed fantastical. Most farmers in the 1600s probably used wooden pitchforks and plows even. Native American people lived much the same way, and some Native American cities, especially those in Central America prior to their conquest, would have seemed impressive to the English rather than the other way around. London itself was a large city, but few of the buildings there would have defied belief or expectations. Meetinghouses and longhouses could be very large and house dozens or hundreds of people, and impressive walled cities housing over a thousand people were not uncommon in North America in the early contact period.

At any rate, this kind of treatment would continue for a few decades, but as the rivalry between the Spanish, English, and French grew over control of the New World, the Native American people took on a new importance - as allies in the many conflicts between the great European powers, as gate keepers to the many resources in North America, and as important trade allies for the colonists. In the 17th century through most of the 18th century, many of the Native American people traveling to England did so as gusts or diplomats, seeking to strike trade and military deals with the English Kings. However, most of the time these Native American people were still being used by opportunistic British people as tools.

In 1729, Sir Alexander Cumings, a minor Scottish noble who had been unable to secure any position in Great Britain, left the Isles to seek riches in the new world. After engaging in various schemes in Carolina by embellishing his wealth, issuing false credit, and defrauding people of their specie, he set out to exploit the Cherokee people in the Carolinas. Sir Cumings, taking advantage of anxieties between the Cherokee and colonists, convinced many of the Cherokee chiefs to pledge loyalty to the British crown. In Tellico, he met a powerful chief named Moitoi who told Cumings that he was supposed to be the emperor of the Cherokee but that political infighting had prevented it. Given that there was never an emperor of the Cherokee people before, it seems as likely that Moitoi was exploiting Cumings role as a power broker for his own advantage. Regardless, Cumings, devoid of any royal power, crowned Moitoi emperor of the Cherokee at Naquasse, and asked Moitoi and six other Cherokee delegates to accompany him back to England. Several months later they appeared in London, much to the surprise of the British policy makers who had no reason to expect a delegation of Cherokee people. The chiefs were instant celebrities who were treated like true royalty on their tour of England. Despite the British government’s unpreparedness, a treaty was quickly drafted and signed between the Cherokee and British which was modeled after treaties with the Iroquois nations, and recognized the royal status of Moitoi. The title of emperor of the Cherokee was passed down hereditarily and carried some clout, though it’s actual power among Cherokee people was questionable. Cumings came away largely empty handed from the affair. By the time of the treaty his reputation in Carolina had caught up with him in London. In the end, he was marginalized and given nothing from the crown or the Cherokee people beyond the money he had schemed away. After failing in other schemes, he was sent to debtor’s prison in 1755, and then committed to a hospital in 1765, where he died in 1775.

These kinds of incidents of course prompted more people to try to repeat the feat. In the aftermath of the Anglo-Cherokee war in 1761, three Cherokee chiefs gained an audience with King George III with the assistance of Henry Timberlake, a Virginia born cartographer who had charted their land after the fighting. The three chiefs were also treated like celebrities and showered with gifts everywhere they went. Timberlake fared somewhat better than Cumings in London, though in the end they met a similar fate. He was promoted to a Lieutenant of the 42nd foot, a Royal Highlands regiment, for his service though he was quickly remanded to half pay. Two years later he accompanied five more Cherokee chiefs to London over a grievance regarding the Proclamation line of 1763. This time the chiefs were not well received, and the trip would be his undoing. In London, he was accused of exploiting the Cherokee for profit and thrown in jail for failing to pay debts, where he died a few months later.

However, after the American Revolution it seems the British, and other Europeans, tired of the whole business. Other people tried it, William Augusts Bowles, most famously, and they were treated cool by the British. The exciting sheen had worn off of meeting Native American people, and the alliances in the New World were less useful than they had been. Where there had once been amorphous boundaries between the British, Spanish, French, various colonies, and Native Americans up to the 1730s, the treaties of Paris in 1763 and 1789 made such separations crystal clear. During Cumings time, colonists were often forced to negotiate with Native American people over trading, fishing rights, or land purchases, which often involved cross-cultural understanding and people capable of navigating both worlds. But as room for ambiguities became smaller and smaller, the expectation from Native Americans was that they would negotiate with the individual colonies, not the British crown directly. The Royal of Proclamation of 1763 generally curtailed colonial expansion into Native American land by drawing a line between European and Native American land and people. It also fueled the antagonism between the British crown and the colonists and contributed to the American Revolution, the outcome of which further destroyed the negotiated spaces between people of European descent and Native American people. By the time of Bowles, Americans rarely negotiated in good faith with Native Americans for their land or bothered navigating their social world for trading contracts.

Later in the 19th and 20th century, Native American people would continue to travel, mostly as entertainers like in Buffalo Bill's Wild West Show.

Further reading:

Definitely take a dive into Beyond the Spectacle. Its a web project that documents and maps every known visit from a Native American to the UK. Here's a direct link to the map, but you should look through everything: The Map

Coastal Encounters: The Transformation of the Gulf South in the Eighteenth Century

Unfortunately, a lot of this comes from my own original research in an unpublished paper, but I am happy to share more if needed.

The first Indigenous peoples came to England in the reign of Henry VII and became part of the court as more of an exotic curiosity. The arrival of three Inuit men with Bristol fisherman from Newfoundland is documented in 1500. They were introduced to the king in 1501. In 1503, an ambassador at the court remarked that two of them (the third either returning to Newfoundland or dead) had assimilated so well into English society that they could scarce be distinguished from native Englishmen. What they thought of their experience is unknown.

In 1531, William Hawkins brought a Brazilian man back to England and presented him to Henry VIII, again as a curiosity. He stayed in England for a year, but died on the return voyage to South America.

It is unlikely that Kalicho, the Inuk captive brought to England in 1576, ever met the Queen before his death a couple weeks after arrival. But she did have a post-mortem portrait of him hanging in her privy chambers. The Lutheran theologian Casiodoro de Reina joked to the Wilhelm IV about Kalicho: "They carried away one of the men of that island with them, although a savage creature; he never could be tamed, and so he died. His painted image has restored him almost to life [and] he is preserved in that Queen's court. As I am writing, I can scarcely contain myself without laughing at this Lucianic lie." The woman (Arnaq) and child (Nutaaq) who arrived in England with him died soon after him. The next year eight other Inuit captives were brought to England, but little is known about them.

In 1584, two Carolina Algonquians, Manteo and Wanchese, came to England and lived at Durham House with Sir Walter Ralegh. We know they appeared before Parliament and at court attired in English clothes and having learned some English, and that Ralegh brought potential investors to his home to meet them, but it is not certain that they met the Queen in person. Both Manteo and Wanchese were elites in their respective communities and they came to England voluntarily as ambassadors, intelligencers, and interpreters, moreso than mere curiosities. Wanchese was quite unhappy with this experience in England, and when he returned to the Outer Banks in 1585 with the Roanoke colonists, he wanted minimal/no contact with the English. After the assassination of chief Wingina/Pemisapan, Wanchese apparently led the attack on fifteen soldiers Sir Richard Grenville had left at Roanoke Island after the rest of the colonists abandoned it in 1586. Manteo, on the other hand, remained a friend of the English and converted to Christianity. He and another Algonquian named Towaye came back to England in 1586 and returned to the Outer Banks with John White's colony in 1587. He was christened and dubbed "Lord of Roanoke and Dasamunkepeuc" because the English regarded him as a trusted ally and subject of the Crown. A fourth Algonquian came to England in 1586 as a captive of Sir Richard Grenville.

Another ten or so Amerindians from Trinidad and Guiana came to England between 1594 and 1596 as part of Sir Walter Ralegh's expeditions in South America. The names of three are John Provost, Anthony Canabre, and Leonard Ragapo, all of them apparently Christian converts, and another just called Harry. There is no evidence they ever met the Queen, especially since Ralegh was in disgrace until 1598, but several of these South Americans formed quite strong bonds with Ralegh and his affiliates. Anthony Canabre spent 14 years in England before returning to Wiapoco as part of Robert Harcourt's Guiana colony. Two or three of them even lived for some time in the Tower of London with Ralegh after his conviction for treason in 1603.

But it has to be said that the majority of Indigenous peoples that came to England did so anonymously as captives, and were treated either as zoological exhibits and objects of curiosity, or forced to provide information that could be used to conquer their homelands. The ones who returned home, like the Etchemin Tahenedo and Skidwarres or the Algonquians Wanchese and Nemattanew, often turned on the English as soon as they were safely back with their people.

Sources:

Alden T. Vaughan, Transatlantic Encounters: American Indians in Britain, 1550-1776, "Powhatans Abroad: Virginia Indians in England," and "Sir Walter Ralegh's Indian Interpreters"