The title of the episode is a play on the 2011 film (and its sequel) The Best Exotic Marigold Hotel, about a group of retirees who purchase a hotel in India. Nanites are of course microsopic nanotechnological robots which were first introduced into the Star Trek universe in the TNG episode "Evolution", although chronologically they have existed as far back as DIS: "Perpetual Infinity".
The Stardate is 59393.7. Cerritos is at the Cosmic Duchess, an interstellar cruise ship with multiple domed biomes which is undergoing a nanite cluster infestation. Boimler's facial hair is increasing, now with stubble on his chin and cheeks. He notes that one of the casinos has Dixon Hill slot machines, Dixon Hill being a 20th-Century hard boiled detective character Picard was a fan of (TNG: "The Big Goodbye"). Mariner's shocked reaction to Jennifer's kiss is understandable, since we were led to understand they broke up in LD: "Trusted Sources".
Jet Manhaver is a background character who was last featured in LD: "We'll Always Have Tom Paris". The way he describes Ransom's using him as fodder is reminiscent of the jokes about the fates of redshirts in TOS.
T'Lyn suggests Jennifer has a brain parasite. The last time a brain parasite caused amorous complications was in LD: "Cupid's Errant Arrow". Jennifer says she is being transferred to the USS Manitoba, which from the name might be a Parliament-class starship (the others we know of being Toronto and Vancouver).
The Risa Bar is named of course after the pleasure planet (TNG: “Captain’s Holiday”) and the biome actually reminds me of the Risa instance in Star Trek Online. This is the first time we’ve actually seen Gallamites although the species was first mentioned in DS9: “The Maquis, Part I” when Jadzia Dax mentioned she had a dinner date with the Gallamite Captain Boday, and Kira evinced some discomfort with his transparent skull. Kreetassans, an easily offended species, first appeared in ENT: “Vox Sola” and consider eating a taboo activity not to be done in front of others. The huge dangerous mountain pointed out by Ransom vaguely resembles the Paramount logo if you squint (it may just be a coincidence).
T’Lyn makes note of the poster of Krog on the Rocks, who performs on Vibe Tubes. Named in this episode, the Vibe Tubes are a futuristic musical instrument, the prop first seen being played in a holographic recreation in TNG: "We'll Always Have Paris".
Admiral Milius’s acolyte is Denobulan, the same species as Dr Phlox of ENT. Denobulan females emit powerful pheremones during mating (ENT: “Dear Doctor”). “Not everyone keeps their genitals in the same place,” is a lesson we first learned in ST VI.
Tendi makes reference to Mariner's parents, who are often separated for long periods of time since Carol Freeman is a starship Captain and Alonzo is a Starfleet Admiral.
Milius' appearance resembles that of Marlon Brando's Colonel Kurtz in the 1979 movie Apocalypse Now, which in turn was written by John Milius, Francis Coppola and Michael Herr inspired by Joseph Conrad's novella Heart of Darkness. In both stories the Kurtz character sets himself up as a demigod worshipped by natives.
An iscosahedron is a 20-sided polyhedron, best known to tabletop roleplayers as a d20 die. Gormaganders (DIS: "Magic to Make the Sanest Man Go Mad") and the "Galaxy's Child" thing (TNG: "Galaxy's Child"), are examples of cosmozoans, giant organisms that live in space.
Captain Tersal says that her parallel universe Endeavor has been through a "month of hell", an allusion to VOY: "Year of Hell". Although Endeavor's scale is due to her being from a universe that is tiny, in the Prime universe Defiant was once shrunk by a subspace anomaly (DS9: "One Little Ship") and Voyager was turned into a Christmas tree ornament by Q (VOY: "Death Wish").