When the boys are playing poker, Danny lays down his cards saying "Full House." Then, Jackson lays down his cards saying "Fuller House."
The scene where D.J. tells the boys how much they remind her of their dad is a reference to a scene in the Full House episode Goodbye Mr. Bear where Danny tells the girls about how much they remind him of their mother Pam.
When Max says, "I'm king of the world!" this is a reference to Leonardo DiCaprio's line in the popular 1997 movie "Titanic".