As far as I can tell, there is nothing in the chapter that tells us what date Fudge is born on. We cannot really even tell what season of the year it is.
But we can tell what day of the week the birthday party is held on. You can find that answer on the second page of Chapter 5. On that page, there is enough information to tell us that the party is held on a Saturday. We can tell the party is on a Saturday because Fudge and Peter's dad can't come. The book says that "He had a Saturday business appointment." That implies to me that the party is being held on a Saturday.