What is the ruling on a mother and her son whispering to each other in a low voice in the presence of his wife, and is this considered a mistake on their part that warrants the wife's distress and withdrawal in this manner?

The Prophet, peace and blessings be upon him, forbade two people from private conversation to the exclusion of a third, because it causes sadness to that third person. This applies to lowering your voices in the room if your normal voices can be heard in the living room. Al-Qurtubi explained that sadness arises from the person's assumption that the conversation is about him in a way he dislikes, or that they belittle him.

Your wife's reaction was wrong despite the validity of her objection to the private conversation. You must now deal with her wisely and apologize to her, promising not to repeat it. It is preferable to tell her the topic of the conversation to relieve her anxiety. Both of you should also comfort the mother and remove any feelings of alienation she may have.