You Do Need power plug adapter while traveling from Japan to Faroe Islands.

Here is Why You Need An Adapter

The outlet and plug type used in Faroe Islands are C, E, F and K and in Japan are A and B. This is how they look like.

Japan

In Japan, outlets of type A and B are in use.

Type A

Type B

Faroe Islands

In Faroe Islands, outlets of type C, E, F and K are in use.

Type C

Type E

Type F

Type K