You Do Need power plug adapter while traveling from Nauru to Falkland Islands.

Here is Why You Need An Adapter

The outlet and plug type used in Falkland Islands is G and in Nauru is I. This is how they look like.

Nauru

In Nauru, outlets of type I are in use.

Type I

Falkland Islands

In Falkland Islands, outlets of type G are in use.

Type G