It turns out that if I'm thinking about the sea nymph in the "Odyssey," there is a common origin to the two words, but if I'm thinking of the West Indian song, it's not clear how the two words are related.
Online Etymology Dictionary says, of Calypso:
lit. "hidden, hider" (perhaps originally a death goddess) from Gk. kalyptein "to cover, conceal," from PIE *kel- "to cover, conceal, save". The W. Indian type of song is so called from 1934, of unknown origin or connection to the nymph.For apocalypse, the same site says:
late 14c., "revelation, disclosure," from Church L. apocalypsis "revelation," from Gk. apokalyptein "uncover, disclose, reveal," from apo- "from" + kalyptein "to cover, conceal". The Christian end-of-the-world story is part of the revelation in John of Patmos' book "Apokalypsis" (a title rendered into English as "Apocalypse" c.1230 and "Revelations" by Wyclif c.1380). Its general sense in M.E. was "insight, vision; hallucination;" meaning "a cataclysmic event" is modern.After finding a mention of the possibility that the word calypso for the musical form might be a folk etymology, I looked around and came across this suggested origin of the word, at eNotes.
It is thought that the name "calypso" was originally "kaiso," which is now believed to come from Efik "ka isu" 'go on!' and Ibibio "kaa iso" 'continue, go on,' used in urging someone on or in backing a contestant. There is also a Trinidadian term, "cariso" which is used to refer to "old-time" calypsos. The term "calypso" is recorded from the 1930s onwards.