In the scene where Billy calls Danny McGraw to apologize for being a jerk to him in high school, the phone starts ringing before Billy is even done dialing the phone.
I thought that too for a while, but I finally figure out (I've seen Billy Madison MANY MANY times) that the sound is part of the music in the background. It's not really a phone ringing
That's the intro to the song that plays throughout the scene, "Telephone Line" by Electric Light Orchestra. The beginning of that song uses telephone rings, which is why the guy in the song is singing, "Hello, how are you?" since he's making a phone call.
The ringing you hear is the beginning of the song that plays through the entire scene. The song is "Telephone Line" by ELO. Listen to it again, but this time, don't listen to Billy and Danny's conversation, you'll hear the song underneath their convo.