You Do Need power plug adapter while traveling from Fiji 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 Fiji is I. This is how they look like.

Fiji

In Fiji, outlets of type I are in use.

Type I

Faroe Islands

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

Type C

Type E

Type F

Type K