https://archive.org/details/CharacterInsightEp251
As primarily a visual phenomenon based on planets, eclipses are not seen much in the space faring stories of Star Trek. However, one eclipse does serve an important story role in the Voyager episode Ex Post Facto, where Professor Tolen Ren's wife expresses the importance of witnessing such an event a few years prior on their planet Banea.
QUOTE:
I haven't been married for years Tom...he treats me as a friend, a companion...he hasn't treated me as a woman since the day of the eclipse four years ago. He was very excited about the eclipse, don't ask me why.
It later became important for Tom Paris to recall this and other information when he is accused of the Professor's murder and is implanted with false memories of the alleged murder to punish him for these alleged crimes. The eclipse was a turning point in the marriage, and that helped Tuvok discover the motive and proof to exonerate Tom Paris of the charge.
Other than visuals referring to eclipses in the Enterprise theme credits or the visuals of other credits, there are no other references that can be found to such phenomenon. It's just one part of stargazing that has not received a ton of play on Star Trek, as there are many more phenomenon to experience when out there amongst the stars.
Stargazing more generally has certainly been shown in many contexts in Star Trek. We see Captain Archer stargazing on a Risa shore leave, and also Trip investigating a stellar nursery from a telescope on Earth in various Enterprise episodes. Kamin from The Inner Light also builds a telescope, as noted on the episode insight a couple weeks ago. Voyager also showed both humans and aliens stargazing as well, perhaps most notably in the episode Blink of an Eye, where Voyager is spotted by an observatory on an alien world when the ship is stuck in orbit of the planet.
Stargazing and enjoying event like eclipses are all about exploration and discovery of the universe around us, so it makes sense that we see references to the same in Star Trek thanks to the central theme of the show. We might have to wait 7 years in America to see another eclipse, but we have far less time to wait to enjoy new Star Trek, and so we go on discovering and exploring.
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