Do they Christmas in Star Trek?
The answer is a qualified 'Yes'.
Aside from the direct representation of the holiday and assorted 'French-Flavored' decorations and traditions in Star Trek 'Generations' where we see Picard and his 'should have been' family gathered for the occasion. The name of the day is never mentioned, but unless the French of the 24th century exchange gifts with a large heavily decorated tree in the room for Bastille Day, it had to be Christmas.
Other references to almost Any Holiday seems to be lacking except for birthdays and the oft-mentioned but never actually celebrated 'Federation Day'. But since the Federation, and thereby Star Fleet, recognizes all and sundry beliefs, it is easy to imagine that Christmas, and Kwanzaa, and Ramadan, and Chanukah, and Occlusion of Saturn too if you wish to observe it, are all celebrated, or at least recognized, by the Federation.
However, with umpteen cultures represented, if the government allowed every holiday on the books to be officially observed, Every Day would be a holiday for somebody and it could easily get out of hand. So the best thing to do would be to simply not recognize any official holidays but allow a reasonable unofficial observance of whatever the locals want to celebrate. And the same would follow through on individual ships, such as Neelix did on VOYAGER for the various crew from different species. For instance, everybody knows Christmas Day is December 25, right?
Well, unless you ask somebody from the Orthodox Christian Religion known as the Coptic Church. In which case Christmas is December 7. Or maybe you happen to be in Eastern Europe or parts of Asia, where Christmas is in the first week of January.
So whichever calendar you adhere to... Peace. And long life in honor of your Holy Day.