One of the most confusing--but also coolest--aspects of the English language is how easily we can take a word and use it as a different part of speech. Verbs can become nouns, nouns can become adjectives, and so on. In the phrase "Japan earthquake," the noun "Japan" is used as an adjective to describe which specific earthquake is being talked about. A "Japanese earthquake" would be using the adjective Japanese to imply that there are different sorts of earthquakes: Japanese earthquakes being different from American ones. This sounds sort of funny, and it is.
"Japan earthquake" is really just a slightly informal shorthand. In formal English, one ought to say something like "the earthquake in Japan" or "Japan's recent earthquake." But journalism and the internet are changing English, and probably most other languages, to be as short and concise as possible.