"일본의 항공사"

Translation: A Japanese airline

[LiKenun]

Japanese airline

[Komerinam]

I think these are the correct translations:
일본 항공사 → Japanese airline
일본의 항공사 → Japan's airline.

[SeanFogart4]

Japan has more than one airline. The possessive includes the properties of the definite article, as do numbers and such, IMHO. So I'd add an s: (one of) Japan's airlines, but they're effectively indistinguishable from Japanese airlines in English AFAIK.