Stardew Valley created by ConcernedApe.
Images used for educational purposes only.

To trigger this quest you need to complete your first year on the farm. Hit the 6th of Spring and Emily will send you a request in your mailbox asking for an Apricot. This is not an easy task.

Apricots grow on Apricot Trees, and to grow an Apricot Tree you’ll need an Apricot Sapling from Pierre’s General Store. Apricot Saplings cost 2,000 gold apiece, take 28 days to grow, must be placed with coverage on all sides (in other words, the eight squares surrounding the Tree must be clear), and only bear fruit in Spring.

In other words, unless you already have an Apricot Tree, Emily’s not getting her Apricot during the second season. Alas. Once you do have a full tree, though, you need only wait until it bears Apricots, which will happen every day. Snag one and keep it for Emily. She’ll give you 600 gold for the Apricot via your Journal.