It's difficult to handle quarrels in public and we always want to maintain privacy and decorum, in such a scenario we always prefer to have meeting room discussions instead of on-the-floor discussions.
I do remember when my manager once wanted to have a discussion and I told him that I am not comfortable doing it here, in such a scenario meeting rooms are always the preferred places to have discussions.
Also, I have seen when a couple argues they argue in the bedroom instead of having it in front of everyone. It's quite obvious that everyone like their privacy and everyone wants to have a discussion in a separate way so that it'll not disturb their integrity.
The general idea behind this is that we all have some comfort zone to have a conversation where we are comfortable when someone touches upon a topic that we are not comfortable talking about when we are in a public place we always insist can we go inside to discuss.
It's a good way to deal with situations where we are not comfortable discussing in an open environment and it results in a fruitful and healthy discussion.