Does the argument has been made between these two person or with another third person?
Not be on speaking terms=not be prepared to be friendly towards each other, usually because of an argument
Idioms & Phrasal Verbs, Gairns & Redman
A daughter asked her mother if she could go on a trip for the weekend with some friends. " Her daughter got very angry and called her mother a nasty name. The mother told her that she was grounded and could not use her smartphone for a week.
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A daughter asked her mother if she could go on a trip for the weekend with some friends. Her mother does not like these friends and said "no." Her daughter got very angry and called her mother a nasty name. The mother told her that she was grounded and could not use her smartphone for a week. The daughter got very very upset (etc.)
Now they are not on speaking terms.
Was the argument between these two persons? Yes, between these two.
not be on speaking terms=not be prepared to even speak to each other.