Learn Dutch: Wij maken nooit ruzie in de groep. - We never quarrel in the group.

You absolutely can say Wij maken nooit ruzie met elkaar (We never argue with each other). However, because the phrase ruzie maken (to quarrel) already implies a conflict between people, you can safely leave out met elkaar when the context makes it obvious.

How does the phrase 'ruzie maken' work exactly?

It translates literally as "to make quarrel" or "to make an argument." In English you "have" an argument, but in Dutch you "make" one. It functions as a fixed expression where maken is the verb and ruzie is the noun.

Why does 'nooit' (never) come right before 'ruzie'?

Time words like nooit generally come before the direct object if that object is non-specific (meaning it doesn't have words like de, het, or mijn in front of it). Since ruzie is a general concept here, nooit is placed directly in front of it.

If I wanted to add 'met elkaar', where would it go in the sentence?

If you add it, it normally goes right after the noun. The sentence would become: Wij maken nooit ruzie met elkaar in de groep.