Which one of these is the correct usage:
1) Your favourite Japanese restaurant
2) Your favourite japanese restaurant (being an adjective in this case, it should be in lower case)
Generally speaking, nations and nationalities are capitalized. This is always the case with things, like cuisine or history, that are closely associated with the the country. Thus Japanese cuisine (not japanese cuisine) and Chinese dynastic history (not chinese dynastic history).
There are a small number of exceptions, when the item described has a more remote connection. Thus
We'll use the good china
(not "the good China") to describe porcelain or vitreous dinnerware. And
(not Japanned furniture) to describe black lacquered furniture. Opinion is divided on whether to capitalize english to describe the spin of a ball.
Rule 2. Capitalize proper nouns—and adjectives derived from proper nouns.
the Golden Gate Bridge
the Grand Canyon
a Russian song
a Shakespearean sonnet
a Freudian slip
In any Grammar book it shall be clear that sentence #1 – "Your favourite Japanese restaurant" – is the correct usage.
According to guidelines in the Chicago Manual of Style, the point of capitalising an adjective is to convey the literal relationship to the proper root noun. So, if you're trying to say that the food comes from there, or the restaurant is in Japan, or is staffed by Japanese persons, or is owned by a Japanese owner, or anything that literally ties it to Japan in a material way, it's appropriate to capitalise the adjective. If you're simply describing the style of the cuisine, a lower-case adjective is acceptable.