Is there a term or expression for:
Two people argue with each other, they don't want to talk with each other any more, so they avoid looking at each other.
What is the common everyday term for it, for example, "we hate each other"?
Not "we hate each other". Hate just suggests that one strongly dislikes another. You can hate someone, but still argue with them face to face, for example.
One expression is (to be) not on speaking terms, or we are not on speaking terms:
not be on speaking terms
If you are not on speaking terms with someone, you refuse to speak to them because you are angry with them:
• They had an argument last night and now they're not on speaking terms (with each other).
be not speaking/not be on speaking terms
if two people are not speaking, they do not talk to each other, usually because they have argued
He was not on speaking terms with his brother or sisters.
be (not) on speaking terms
To have an amicable, although perhaps guarded or superficial, relationship with someone. This phrase is often used in the negative to show that two people are estranged.
It took a long time, but my ex-husband and I are finally on speaking terms these days. After that argument last night, I'm not on speaking terms with Stephanie.
I don't know Kyle that well, but we're on speaking terms, and he seems nice enough.
be on ˈspeaking terms (with somebody)
2 (also be ˈspeaking (to somebody)) be talking to each other again after an argument: Tony and Craig had a big row and are not on speaking terms. ♢ You’re lucky I’m still speaking to you after what you did!
It doesn't explicitly mean that they avoid looking at each other, but I think we can imagine that to be the case in certain contexts.
After two people have argued, and they are friendly, or somewhat friendly, again, then you say they are on speaking terms.
The expressions be speaking and be not speaking are synonyms with the above, as noted in the Longman and TFD entries, and in the OALD.