I've never seen the movie myself, but from what I understand there isn't any dialogue saying it's the Illiad. He gives her a book and she says it's a signed first edition without either acknowledging what book it is, but you can tell from the cover. My guess is the props department just gave them a random book without considering the book they gave them doesn't make sense in context.

A lot of hollywood writers and especially script doctors aren't really well versed in literature or ancient world classics, it's not germane to the form they produce. I know a hollywood writer and he certainly didn't graduate with an MA in English.