Yes, you can. There is no rule saying how far in advance, but it's pretty common to wish people "a Merry Christmas and a Happy New Year" together, so you'll sometimes hear it a week or more in advance. After Christmas, you'll hear it all the time leading up to New Year's day.
Generally if it's close to Christmas, you can also offer New Year's greetings.
As for after the holiday, again there's no rule, but you'll hear it for at least several days, maybe up to a week.
You won't hear it for as nearly as long after January 1st as you're likely to hear it before January 1st, however.
As long as you're not saying it in July, people won't care.
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