I think you have to keep trading with the girl and it will appear after you reach a certain relationship level.

The neighbor islands are unlocked when you clear the area above the main level (notice that table in the trees?). That will open up the other islands where you can trade and buy goods too.

I got one blue map piece as a prize, the other was in a boat trade. They are few and far between.The gems I got have been through boat trades as well.

You can pay 1 diamond to take the item off the board, or if you wait long enough (several hours), Wesley will buy it. At least that is what happened to me.