The climax of Breaking Dawn is when the Volturi come to the forest clearing to charge the Cullens with breaking one of their laws: creating an immortal child. Of course, the Cullens saw this coming, thanks to Alice, and they banded together a crew of fellow vampires (and werewolves) to testify that the child is in fact not immortal, but rather born from a human.
Though the scene in the book doesn't escalate to a full-fledged battle of any sort, the plotline of the novel certainly leads up to this moment, which is why it would be considered the climax. The conflict that arises within the novel is Bella's pregnancy—one that nearly kills her—but the child is born, Bella survives, and everything seems to be fine until the Volturi find out. Bella even goes as far as to obtain passports for Renesmee and Jacob so they can escape, as she fears that the Cullens aren't going to win against the Volturi.
The conflict finally comes to its peak when the Volturi appear, the Cullens (and witnesses) prove that they haven't broken any laws, and the conflict resolves so that the Cullens are able to live peacefully as the novel ends.