Kia orana all,
can someone help with my quandary.
For those who don't know, the Cook Islands embody Fifteen Islands each with their own character and proverbs. Is it correct to say: Cook Island Proverbs, or: Cook Islands Proverbs. Much appreciated.

I'd venture it could be either, or both, depending on whether you're speaking of..

Cook Islands proverbs

or

Cook Island proverbs

-joe (dithering) friday

Excuse the ambiguity. The statements refer to Proverbs that come from the Cook Islands. Based on your Two suggestions, The first seems correct. But would Cook Island Proverbs be correct also?

I'd go with Cook Islands Proverbs, so as not to confuse the proverbs of the Cook Islands with the proverbs of Cook Island (which is not a part of the Cook Islands and is uninhabited so there are probably few proverbs to be gathered). You could reword the title to Proverbs of the Cook Islands.

Yes thanks, that makes sense.

I'd say "Cook Island proverbs". "Island" can be an adjective (as in "an island nation"), so why can't "Cook Island" be an adjectival phrase modifying "proverbs"?

It can, but, as pointed out above by Nuncle zmjezhd and by tsuwm, it would be misleading at best and impossible at worst.