You Do Need power plug adapter while traveling from Saint Barthélemy to Norfolk Island.

Here is Why You Need An Adapter

The outlet and plug type used in Norfolk Island is I and in Saint Barthélemy are C and E. This is how they look like.

Saint Barthélemy

In Saint Barthélemy, outlets of type C and E are in use.

Type C

Type E

Norfolk Island

In Norfolk Island, outlets of type I are in use.

Type I