Can a moose and an elk mate and produce offspring?

It depends on whether you ask an Englishman or an American.

In British English, an "elk" is the animal with the Latin name "Alces alces". The word is old and comes from Proto-Germanic. The animal is found in both North America and Northern Eurasia. When European settlers in North America came across the animal "Cervus canadensis", they thought it looked similar so they called it "elk". It's in fact a large deer found in North America and Eastern Asia, but not in Europe. The name "elk" stuck, so a new American word for Alces alces was borrowed from Native American languages: "moose".

So, to an Englishman, "elk" and "moose" are the English and American names for same animal. To an American, they are different species that, from what I gather, are not closely related enough to produce offspring.

https://en.wikipedia.org/wiki/Elk
https://en.wikipedia.org/wiki/Moose


(Edited: spelling mistakes. Edited again: clarity)

I found the following curious entry in a science journal. No photos, unfortunately.

An Alleged Moose-Wapiti Hybrid in Montana.—California Fish and Game (vol. 17, p. 198; 1931) contains a reference to what is said to be the first known specimen of a ‘moose-elk’ or in our nomenclature a ‘moose-wapiti’ hybrid. Known to the United States forest rangers as “the elk with the funny horns” this curious cross was recently killed in the Deerlodge National Forest, Montana. The rangers had for the past five years known of the animal, which associated and grazed with the wapiti, but the horns and the body of which were half moose and half wapiti. When first seen in 1925 he appeared to be about a three-year-old, and his weight when slain was 1100 pounds.

http://www.nature.com/articles/128676a0