Okay, I found a little info on the subject:
First off, it seems that we have no names for these deities, save for one exception. The two possibilities for why this may be, from what I've gathered, are: a. They didn't have personal names; rather, they were simply called by what they were, but in the languages of the Floridians - Sun, Deer, etc. b. They had personal names but these were lost through time. There are only about ten recorded words from the Calusa language, for example. Any missionaries, explorers or conquerors who recorded some aspects of their religions didn't give any names for these deities - either they didn't know the names, or perhaps this was a religious ploy to rid the world of these deities which they perceived to be false idols. The one exception may be the name of the sun, but I don't know if it's the personal name of the sun or simply the word they used for the word 'sun', and it might have even meant neither but something like "powerful being" which was eventually attached to the sun. According to "Why Have You Come here?: The Jesuits and the First Evangelization of Native America", the sun may have been called "Ate" because of this exchange between the missionary Father Juan Rogel and one of the Caciques (chiefs):
"Rogel: Do you believe in the unity of God, the creator of the universe?
When Rogel used the word "God", the interpreter probably translated it with the Indian word "Ate". To the Indian the Ate was the sun, a superpower, one who was responsible for the light, the brightness..."
In "Soldiers of God: The Jesuits in Colonial America, 1565-1767", pg. 33, it says of the Floridian tribes in general:
"Three deities ruled over nature, political matters, and warfare. The sun was the most honored deity and fire was its symbol."
There was a merchant from colonial Jamaica named Jonathan Dickinson who was captured and kept in captivity for a certain of time by Floridian tribes. I wasn't able to understand which of the tribes held him captive. On Wikipedia it says the Jobe (Jeaga), on this essay it seems to say the Tequesta and here it says he was with the Jeaga and Ais.1
According to this article, they would worship "spirits and constellations" and it's possible that the sacrificing of sharks was involved. In this essay its says that Dickenson said that their religion involved worshipping the sun and moon.
In the same essay, pg. 18-19, it says that Carlos, one of the chiefs of the Tequesta, even after allowing the erection of a cross in his village, refused to remove certain idols and even ordered that a Spaniard be sacrificed every year to the devil.2
Here it says:
"According to some accounts, when a Tequesta chief died the leader's body and small bones would be buried while the larger bones would be separated and placed in a box for the Tequesta people to worship as gods."
And according to "Race, Nation, and Religion in the Americas", pg. 39, the Caciques (chiefs) were considered to be the descendants of a group of deities, chief among them the sun.
In this essay, pg. 18, it says that they, too, worshipped the sun and moon:
"The Caloosas of slightly north of this area held their principal festival at the first corn-planting. A deer was sacrificed to the sun and its body elevated on a pole for religious veneration."
In "Exploration of Ancient Key-dweller Remains on the Gulf Coast of Florida", pg. 64-65, it says:
"...according to the accounts given by more than one writer on Florida, the deer must have been regarded among some of the Floridian tribes as one of the gods of the day or of the dawn...In such event they symbolized...the four sets of the sun's rays that are supposed to correspond to the four quarters of the world, as well as to the four sets of three months in the corresponding four season of the year over which the sun god is believed to have dominion - since he creates all the days thereof."
Here, pg. 30, it says that at some point they had two idols:
"...the principal one, "a board sheathed in deerskin with its poorly formed image of a fish that looks like the barracuda and the other a figure like tongues." The other idol is described as the god of the cemetery, "a head of a bird, sculpted in pine,"..."
In "The Evolution of the Calusa", pg. 6, it says:
"According to Father Juan Rogel..the Calusa pantheon included three deities who governed the world. These deities were ranked hierarchically, the first deity being the greatest. This deity governed or pertained to phenomena of nature, such as movements of the stars and weather. The second deity pertained to political matters, such as the rule and government of the region. The third, and least important, deity pertained to warfare. The hierarchical nature of the Calusa pantheon, however, may have been a result of the misconception of Rogel.
Here it expands a little bit on the first deity by saying:
"Each god had a specific role that they were responsible for. The most powerful was in charge of the moon, stars, sun, and weather..."
On Enotes it says:
"Like most Southeastern Indian people, their religion contained what some ethnohistorians call the "Southern Cult." While their actual practices are not all that clear, there are many icons, including skulls, circles, and various spiritual figures, that Florida Indians have in common with Mississippian culture. Their society was highly stratified, and it is known that they venerated ancestors, as many Indian people did."
According to this essay, pg. 37, the Ais had a moon ceremony. No name is given for the moon.
According to this essay, pg. 36, the Jeaga had a moon ceremony. No name is given for the moon.
1 Dickinson's full account of his time as being taken captive by the Floridians can be read here.
2 From the accounts of missionaries such as Rogel, it seems that the Native Americans, or at least the Floridians, had great difficulty grasping the concept of the Christian one god and the Christian concept of any other divine force being "of the devil". The main reason seems to because they believed that man has three souls (plural, as opposed to the Christian oneness). The missionaries' description of Carlos sacrificing "to the devil" appears to not really mean "the devil" in the Christian sense of the word, at least from the Floridians' point of view, but to another, non-Christian deity. I believe this is comparable to other Native American tribes who had trickster gods and other ethnic groups in general who had gods of things that Christianity would term as "evil", such hell/the underworld. While in Christianity such a being would not be considered a god but a dark force that must be eradicated, in these other religions even the "bad" gods are part of the pantheon.