What country is reputed to have issued the first postage stamp to commemorate Christmas, in 1937?

Actually, it is Canada in 1898. The stamp is the 1898 Canada two-cent with the Mercator map. (Gerhardus Mercator was the most notable geographer of his time, and his world map of 1569 won lasting fame.) Most often called the 'Map' stamp or the Imperial Penny Postage issue, the stamp also gets credit for being the first ever Christmas stamp.

The next nation with Christmas stamps was Austria in 1937 with two stamps referred to in the Stanley Gibbons catalogue as Christmas Greeting Stamps. The stamps feature a rose and zodiac signs. Brazil issued four semi-charity stamps in 1939 depicting the three Kings and the star, an angel and child, a southern cross and child, and mother and child. Hungary gets next in line with a Soldier's Christmas semi-postal depicting a soldier and emblem (1941).

http://www.psestamp.com/articles/article1087.chtml

The Canadian stamp when it was officially released on December 7, 1898, bore, not only Mercator's map, but also the words "XMAS 1898."

"Reputed" is a good word to use.

The Canadian stamp is interesting, but the only thing is has to do with Christmas is that a new postage rate went into effect that day.
https://en.wikipedia.org/wiki/Christmas_stamp

This Christmas Philatelic Club page explores the controversy and proposes that Hungary issued the first expressly Christmas stamp in 1943. It says that the Austrian stamp of 1937, which I've been unable to find a picture of, was to be used for sending birthday greetings.
hwcn.org/link/cpc/1rstxmas.html webpage no longer exists