From the CT State Library page on state nicknames- 

According to the book State Names, Flags, Seals, Songs, Birds, Flowers, and Other Symbols by George Earlie Shankle (New York: H.W. Wilson Company, 1941):

“The sobriquet, the Nutmeg State, is applied to Connecticut because its early inhabitants had the reputation of being so ingenious and shrewd that they were able to make and sell wooden nutmegs. Sam Slick (Judge Halliburton) seems to be the originator of this story. Some claim that wooden nutmegs were actually sold, but they do not give either the time or the place.”

Yankee peddlers from Connecticut sold nutmegs, and an alternative story is that:

“Unknowing buyers may have failed to grate nutmegs, thinking they had to be cracked like a walnut. Nutmegs are wood, and bounce when struck. If southern customers did not grate them, they may very well have accused the Yankees of selling useless “wooden” nutmegs, unaware that they wear down to a pungent powder to season pies and breads.” Elizabeth Abbe, Librarian, the Connecticut Historical Society; Connecticut Magazine, April 1980.

But, according to this article,  “Early sailors would bring the valuable seed back on their foreign voyages. “