What an excellent and interesting question! As it would be impossible to treat the entirety of the Conquest in any detailed way in a single comment, I endeavor to offer you a broad sketch of the relationship between the conquerors and their favorite literary genre. As such, feel free to ask any questions in a reply and I will do well to answer them.

The seminal study on this subject is Irving Leonard's 1949 monograph, Books of the Brave. More recently however, other scholars such as Jennifer Goodman and Victoria Muñoz have treated the topic in their own works (respectively, Chivalry and Exploration from 1998 and Spanish Romance in the Battle for Global Supremacy from 2020). Our point of departure will here be the vogue that romances of chivalry enjoyed in Spain. Garci Rodríguez de Montalvo is usually credited with the boom and popularity of chivalresque works in sixteenth-century Spain. Shortly following the conquest of Granada and the end of the (re)Conquista of the Iberian Peninsula, Montalvo edited and added two books to the already popular romance Amadís de Gaula. This wonderful book is considered by some scholars to have been a venerable best-seller, and there are several testimonies to its immense popularity: not only did the Amadís saga eventually span 12 books written by various authors, but it also penetrated society from the bottom up. As Noel Fallows discusses in his book Jousting in Medieval and Renaissance Iberia, royal festivities often took inspiration from the Amadís and its sequels. That is, the competitors would dress as characters from the romance - Phillip II included - or the events themselves would center around adventures extracted from the text (see pp. 48-50 and the discussion of the Tournament of the Dark Castle).

The Amadís saga proves to be useful in talking about the influence of romances on the conquistadores because they themselves reference it. In his eyewitness account of the conquest, Bernal Díaz del Castillo describes the architecture of Mexico thus: "And as we saw so many cities and inhabited villages in the water, and many other large settlements on land, and the road that led to Mexico, we were stupefied ['admirados'], and we said that it all these things seemed like the enchantments recounted in the book of Amadís" (trans. and emphasis mine, ch. LXXXVII). Hernán Cortés's secretary, Francisco López de Gómara, describes the rapids of Mexico by referencing the Amadís as well: "That crossing was a flat crag, as long as the river was wide, upon which the water fell without covering it; and although it seems fabulous or like one of the enchantments of the Amadís, it is most true" (trans. mine). It is from the Amadís saga as well, for example, that we get the name of the state California. In the fifth book of the series, the Sergas de Esplandián, we are told that "on the right-hand side of the Indies there was an island called California, close to the Earthly Paradise, which was inhabited entirely by black women without there being a single man among them, for they lived in the manner of the Amazons (trans. mine, ch. CLVII).

Other romances feature in the writings of the conquistadores, such as those touching upon the Matter of France. Particularly popular among them, due to the recent ascension of Charles V to the throne, was Nicolás de Piemonte's Historia del emperador Carlomagno y los doce pares de Francia. Hernán Cortés, for example, is recorded as saying "God give us the same good fortune in fighting as he gave to the Paladin Roland" when he made landfall in Mexico. Roland, of course, is the chief of Charlemagne's Twelve Peers. There is also the case of Gonzalo Fernández de Oviedo, who is credited with writing the first American novel; in reality, he wrote a chivalric romance titled Don Claribalte. He is a particularly interesting case in that he is also responsible for writing the Historia general y natural de las Indias, a monumental history of the early stages of the Conquest.

To the last part of your question, then, we can say to some degree that the Conquest came to influence the later romances of chivalry. I refer particularly to those written under Phillip II. A good example of this is Diego Ortúñez de Calahorra's Espejo de príncipes y cavalleros (translated into English by Margaret Tyler as the Mirror of Princely Deeds and Knighthood). This romance was dedicated to the son of Hernán Cortés, Martín Cortés, and the hispanist Daniel Eisenberg believes that its author was his tutor (see XVII-XVIII in his introduction to the work). Rather than bearing an influence upon the monsters or adventures of the book, the Conquistadors became a target of polemic for Calahorra. In chapter 44 of book 1 for example, the narrator pauses to reflect upon "codicia insaciable" (limitless greed), which he says is a cause for man's perdition in the world. Such discourses are usually seen in the larger context of invectives against the Conquistadors, such as those launched by Bartolomé de las Casas (see vol. 2 pp. 204 and the accompanying note in Eisenberg's edition of the romance).

This answer is a bit all over the place, but your question can be approached from various angles. Without a doubt, the conquest and the romances of chivalry go hand-in-hand. Their influence, however, is perhaps best examined in isolated cases where one can be go into detail about one thing or another without painting with large, general brushstrokes. I encourage you to consult the bibliography, particularly Goodman's Chivalry and Conquest.

Bibliography:

Calahorra, Diego Ortúñez de. Espejo de príncipes y cavalleros, ed. Daniel Eisenberg. Espasa-Calpe, 1975.

Castillo, Bernal Díaz de. Historia verdadera de la conquista de la Nueva España. Porrúa, 2017.

Fallows, Noel. Jousting in Medieval and Renaissance Iberia. Boydell Press, 2010.

Gómara, Francisco López de. La conquista de México. Historia 16,1987.

Goodman, Jennifer. Chivalry and Exploration 1298-1630. Boydell Press, 1998.

Montalvo, Garci Rodríguez de. Las sergas de Esplandián, ed. Carlos Sainz de la Maza. Castalia, 2003.