No, your child does not need a ticket if their b-day falls during the week you are there. So long as you book the trip to start while they are 2, they are two for the whole trip! Same goes for 9yr olds that turn 10... which is a big savings from a "child" to an "adult"!!
I recommend to all my friends and clients to try and go before a little one turns 3, especially if there are 5 in the family... then they can still get a standard room at most all the resorts (mainly the values).