Apricorns are untradable.
You can pick them from trees once you get the Apricorn Case. (That's obtainable from the guy in the house you first see after you leave Cherrygrove - it's along the way to Mr Pokemon's house). You can get one from a tree a day, and there's 7 kinds of Apricorns. Take them to Kurt in Azalea Town and he'll turn an Apricorn (or multiple of the same kind) into Pokeballs of a certain kind depending on the Apricorn.
You can also use them to make juice, which'll increase the stats of Pokemon for the Pokethlon.
That should answer it, really... Try the Questions and Answers thread next time. Thread closed.