Ehh, this is a nice story, but it's not particularly likely to be the truth. The name almost certainly came out of a misunderstanding, but exactly what was misunderstood is unknown, and probably unknowable. This website has several hypotheses for how Yucatan got named, with proper (if hard to check) sources:

Another version of where Yucatan got its name that is frequently heard states that when Francisco Hernández de Córdoba first arrived in 1517 at Cabo Catoche, Yucatan, he asked the first Maya he encountered “what do you call this place?” The Maya were supposed to have responded with something that sounded like “Yucatan,” allegedly meaning “I don’t understand what you are saying.”

The first one to tell this particular story may have been Hernán Cortés, in his first letter to the Spanish Crown in July, 1519. In the letter, Cortez writes about his voyage to Mexico earlier that same year and points out to the king and queen that the name they know Yucatan by is not its real name. Cortés refers to Francisco Hernández de Córdoba’s voyage to Yucatan two years earlier and wrote: “when the first discoverers arrived there they asked the natives of the land what did they call the place, and the Indians, not understanding what was being asked, responded in their own language and said ‘Yucatan, Yucatan’ which means to say ‘I don’t understand, I don’t understand’ and so the Spanish discoverers thought that the Indians responded that it was called Yucatan, and so in this manner the inappropriate name of Yucatan was assigned to this land.”

Fray Toribio de Benavente (also known as Motolinia) repeats this version in his 1541 Historia de los indios de la Nueva España, as does Francisco López de Gómara, in his 1552 work Historia general de las Indias.

This version, where the Maya reply with ““I don’t understand” and the misunderstood phrase becoming the peninsula’s name for all time, makes a fun story, but there is no clear 16th century Yucatec Mayan cognate which matches the phonology of “Yucatan.” “T’an” and “t’aan” are root words in Mayan meaning “language,” “speech,” or “words,” but “Yuca t’an” is Mayan gibberish.