You Do Need power plug adapter while traveling from Brunei to Solomon Islands.

Here is Why You Need An Adapter

The outlet and plug type used in Solomon Islands are G and I and in Brunei is G. This is how they look like.

Brunei

In Brunei, outlets of type G are in use.

Type G

Solomon Islands

In Solomon Islands, outlets of type G and I are in use.

Type G

Type I