There are quite some difficultires here because of a lack of sources on Native Americans moving to Europe. From what I've read I haven't found anything on indigenous people actually 'settling' to live in Europe, which doesn't mean it didn't happen, against all odds -- to me the idea that someone as 'other' as a native American person could have lived somewhat normally in Early Modern Europe seems highly improbable. Anthony Pagden (in "European Encounters with the New World", pp. 31-33) mentions cases of natives being brought over as 'curiosities' rather than to settle, esp. that of Tainos brought as "gifts" to the Spanish monarchs by Columbus:

What, in fact, happened to the savage as he disappeared into the thickets of European civilization? We have records of very few cases. Most vanished without a trace. The Indians with which Columbus returned from his first voyage, the Tupinamba Motaigne met at Rouen, surface briefly as data used to embellish texts which have only transitory use for them. Columbus loses interest in them when they are declared not to be slaves. [...] These creatures are only ever of interest as representatives of their own worlds. [...] Exposed to Europe, the 'savage' can only sustain his position of detachment for a short while. The critique of our ways which nearly all such visitors express, will lose its force the longer the critic remains absent from the culture which had made it possible at all.

Despite the loaded language, I think these are interesting points regarding the Europeans' limited interest in native Americans in Europe. An example closer to your question is mentioned by Restall (in "Seven Myths of the Spanish Conquest", p. 88). This deals with a native person later christened Martín Pizarro who worked as translator for the conquistador Pizarro in colonial Peru. For his services he was first rewarded with titles and encomiendas, and after participating in Pizarro's revolt traveled to Seville to appeal a sentence against him where he died around 1550. Shortly after, "[i]n 1567 his half -Spanish daughter, dona Francisca Pizarro, was at court in Madrid petitioning the king for a pension as did so many of the descendents of conquistadors". Such petitions in Madrid were common by both Spaniards and mestizos, i.e. people of Spanish and native descent. Nonetheless, these cases don't really count as "settled" as well.

The case that to me comes closest to what you're looking for is not by an indigenous but rather a mestizo person, the famous 'Inca' Garcilaso de la Vega. His is an exceptional case as he descended both from the pre-conquest Inka elite and from Spanish conquistadors. This enabled him to move to Spain with the support of an uncle in 1560, where he sought the acklowledgement of privileges from his father's side. His chronicle "Comentarios Reales de los Incas" proved to be a success in Europe, and de la Vega remained in Spain until his death. As I said this is quite an exceptional case, as the other major native and mestizo authors known to us from the 16th c. were not published in their life-time and remained in the Americas (except for occasional petitions in Madrid, see above).