Not unless your DM allows this sort of thing, which he is under no obligation to do. There are no official rules regarding this situation, which in general means that advancement doesn't happen.
I personally feel that animal companions ought to advance, not because it is particularly realistic, but because it makes for a better story to keep the same companion as long as possible. Heck, I'd encourage my players to advance their companion and awaken it as early as possible so that I could start giving it experience points towards the purchase of class levels.