In Twilight, the first book, Edward does not bite Bella except for when he sucks the venom out of her from James' bite. I don't think that's exactly what you mean though. However, if you continue to read the books, a great conflict that occurs through many of them is that Bella wants Edward to bite her, and in doing so turn her into a vampire. Edward is in opposition to this, and argues with Bella repeatedly about turning her into a vampire. Edward does not wish to turn her, whereas Bella wishes to become a vampire. Edward even offers her a deal, that he will be the one to turn her as she wishes, only if she will marry him. Whether Bella takes that deal, and whether she actually becomes a vampire, I'll leave for you to find out as you read on.