How can I solve a problem between my boss and a staff member who do not speak to each other?
Is it your problem to solve? I dare say their refusal to talk can make the situation tricky and the work atmosphere not the best, but I’m not certain you can sort this out. I think we should assume there is some history between the two of them, an argument, family issue, an affair. Words have probably been said that can’t be unsaid.
Maybe they both need to keep their jobs, and have agreed to differ, come to a situation where they each just get on with life and ignore each other. Your interference may rock the situation so that it deteriorates. So be very careful.
How you address it, if indeed you do that, very much depends on the dynamic and relationship you have with each of them.
If you were their manager I would suggest you tackle each of them separately about it. Point out that it is unprofessional and affecting the morale of the group.
But you are not their manager, so be careful you don’t exacerbate the problem
Read the sections on effective communication before you do anything.
And this one, which will offer some guidelines on feedback.
Who you tackle first depends, as I said earlier, on the relationships between you all. But perhaps a light touch, a comment, rather than a confrontation, may be your best bet. But actually I think it is up to your boss to sort this out.