The indigenous peoples of Cuba were the Taíno, an Arawak people who inhabited most of what is now Cuba, along with other Carribean islands such as Jamaica and Hispaniola. The Taíno have their roots in South America, particularly in what is now the Orinoco Valley in Venezuela, and divided from the Arawak by migrating to the northern Lesser Antilles, and even further into Hispaniola, modern-day Puerto Rico, and Florida. They relied heavily on canoes in trade and transportation; they were accomplished seamen and traveled throughout the Caribbean. Some examples of large canoes found could carry thirty people. There is a strong amount of evidence for contact between the Taíno and the Mayans, mostly through archaeological research. There are two pools of evidence supporting this: Mayan trade goods found in Taíno heritage sites and some indicators of cultural exchange, mostly through the similarity of Mayan and Taíno ballgames. On Columbus' first voyage into Cuba, they found a cask of beeswax in easternmost Cuba, which is most likely to have come from Yucatan; to quote Ronald Canter in his paper 'Yucatan Channel and Trade:' "The common honeybee was unknown in the Americas before contact. Only the stingless bees Meliponini beecheii and M. Yucatanica were suitable for honey production, and the Maya were virtually the only beekeepers. Bartolome de Las Casa, in comments on Columbus’s “Journal of the First Voyage”, noted that the people of Cuba did not keep bees or produce beeswax themselves. Las Casas speculated that the wax had come from a wrecked Maya trading canoe. Given the currents, this seems unlikely, but not impossible."
In addition, recently mineralogist George Harlow traced Preclassic jadeite axes from a site on the Island of Antigua to the mines the material was retrieved from in modern-day Guatemala. It is highly unlikely this material drifted across the Caribbean, which suggests the possibility it was traded indirectly through South/Central American societies. Besides that, there have been observed similarities between Mayan ballgames and the Taíno game known as 'batay', a word which also seems to appear in Classical Mayan inscriptions to refer to the game; similar courts as those found in Mayan sites have been found in the Dominican Republic and Puerto Rico, all more recent than 650 AD.
Sources:
Canter, Ronald 2006 "Yucatan Channel and Trade." FAMSI Journal of the Ancient Americas, 11-28-2006. (PDF warning) http://research.famsi.org/aztlan/uploads/papers/Canter-Yucatan-Channel-and-Trade.pdf
Colombus, Christopher 1493 “Journal of the First Voyage of Columbus.” In American Journeys: a Digital Library and Learning Center, Document No. AJ-062, Wisconsin Historical Society. With notes by Bartolome de Las Casas. http://www.americanjourneys.org/texts.asp
Petit, Charles 2006 “Jade Axes Proof of Vast Ancient Caribbean Network, Experts Say.” National Geographic News, 6-12-06. http://news.nationalgeographic.com/news/2006/06/060612-caribbean.html
Taylor, Dicey 2001 “Taino: Ancient Voyagers of Sea and Spirit.” Precolumbian Society, exhibit/lecture 12-8-01, Philadelphia, PA. http://www.nyboricua.com/comnovpg04.htm
Jones, Chris 2001 “The Ballcourts of Puerto Rico.” Precolumbian Society, lecture 5-12-01, Philadelphia, PA
The Yucatan of the Late Postclassic boasted an extensive circum-peninsular trade network stretching along Mexicos Gulf Coast down into at least Honduras, but evidence that this extended into a trans-Caribbean network is tangential at best. Much is made of the fact that batey, a ball game played throughout the Caribbean, has a strong resemblance to the general pattern of the Mesoamerican ball game played on the mainland, but the route and manner of this connection is unclear.
A FAMSI article covers several archaeological findings which may point to a connection between the Yucatan and Cuba. Unfortunately, some pieces of jadeite of disputed origin, decades old references to potentially Maya pottery sherds, and a single finding of what might be a Taino "vomit ladle," does not paint a convincing picture for any sort of sustained system of trade. Callaghan (2003) suggests, on the basis of computer modeling of potential sea voyages, that if any contact did occur, it was mostly likely that Caribbean voyagers were the ones visiting the Mainland, given that wind and currents would tend to push a sea craft in that direction. Meanwhile, there was little incentive for a Yucatec trader to row into the open sea for a week (under optimal conditions), as opposed to making trips along already established and lucrative routes along the mainland coast.
As for the Aztecs, 1466 CE puts this shortly before the death of the great Motecuhzoma Ilhuicamina, whose conquests of the Totonacs put substantial parts of what is now Veracruz state under Aztec dominion. There's no evidence that I'm aware of that either the Totonacs or the Aztecs were at all interested in launching canoes towards uncertain destinations in the Caribbean. The latter, in fact, had already begun to turn their attention towards extending their rule up along the coast and into West Mexico.