In the book New Moon, Jacob doesn't give Bella a birthday present. He doesn't even spend time with Bella until after Edward leaves her and they begin to repair the motorcycles together. Bella purchases two old motorcycles because when she rides she sees Edward in her mind and it is a "rush."
In the movie, New Moon, Jacob gives Bella a "dream-catcher." This is an Indian relic that supposedly catches the bad dreams and lets only good dreams in. During several scenes in the movie you can see the "dream-catcher" hanging from the head-board on Bella's bed. Charlie, Bella's father, even refers to the dream-catcher during a scene when Bella is having terrible nightmares.
Hope this helps.
New Moon opens with Bella's eighteenth birthday, and she celebrates it with the Cullen family. Jacob is absent from Bella's birthday celebration and, in fact, doesn't appear in the book until several months after Bella's birthday. Jacob's entrance into the novel is marked by Bella's purchase of two motorcycles. She visits Jacob to see if he can help her restore the bikes and offers to give her one of the bikes in exchange for his labor and for riding lessons. In the conversation in which she makes the offer, Bella asks Jacob whether he can legally drive. When she realizes she has missed Jacob's birthday and apologizes, Jacob points out that he missed her birthday as well. They briefly discuss the possibility of having a joint birthday party, but before this happens, Jacob discovers he is a werewolf, which complicates his relationship with Bella. The party never happens, and Jacob technically never gives Bella a gift, unless you consider his restoring the bikes and teaching her how to ride her birthday gift.
Bella spent her 18th birthday with the Cullen and that is when Bella got a cut and Jasper launched himself at her. This is how Edward decides to let Bella go so she could live a normal life without him. AFter this Bella became closer to Jacob and he gave her a present when she graduated.
Bella wasn't really close to Jacob around her 18th birthday. She had spent her 18th birthday with the Cullens, which is when that whole fiasco happened with Bella getting a paper cut and Jasper going crazy to feed from her. This was also why Edward decided to leave Bella in New Moon. Bella becomes close with Jacob a while after this happened. But Jacob did give Bella a present on her graduation day in Eclipse. It was a carved replica of himself in wolf form.