Hi, Bugatti.
What is better to use in every-day English? Fall out or quarrel.
They have slightly different meanings to me. If you tell me that they "fell out with each other" or "had a falling out", I'll understand that they had a big argument about something. They are still angry about it. They might not be speaking to each other now.
If you tell me that they "had a quarrel", I'll understand that they had an argument or a disagreement about something. It might not be serious. They may be perfectly happy after their quarrel.
Here in the U.S., I hear "argument" and "fight" far more often than "quarrel" in the speech of my friends and neighbors. "Quarrel" is widely understood, though, and it's fine to use that word if you want to. People often use "fight" in talk about loud, angry arguments even if no physical fighting was involved.
