Is it permitted to say the word “christmas”?
This is permitted.
Although it is not permitted to say the name of idols and gods, in this case the reference is to the festival alone, and not to any deity, and this does not involve any halachic problem. However, some prefer to use Xmas.
The Shulchan Aruch rules that (Yoreh Deah 147:1) “one who takes a vow in the name of a foreign god receives lashes, and one cannot even mention it by name, with or without a reason.”
However, this prohibition applies to names that imply a deity, and not a simple given name (like Yeishu or Jesus). This principle is noted by the Hagahos Maimonion on the Rambam (Avodah Zarah 5:3), who explains that only names that suggest divinity are forbidden, and this is cited by the Shulchan Aruch (147:3) and more specifically by the Gra.
Therefore, since the festival Christmas does not imply any divinity etc., and it is not even a name of an idol but only of a festival, there is no problem in mentioning it.
Having said this, some prefer to say Xmas.