Microsoft SQL Server login using Active Directory Credentials

The error message is that the certificates were issued by an untrusted authority. That would be your AD domain, most likely.

A couple of workaround that you might want to try:

  1. This post suggests clicking the options button on your connection window above and adding TrustServerCertificate=True under Additional Connection Parameters. This is basically telling SSMS that yes, you know, just do it.
  2. If that doesn't work, this post suggests using runas in order to trick Windows. For example, runas /netonly /user:domain\username "C:\path_to\ssms.exe" I've never done this with SSMS but it's worth a try, since the OP seems to be in a similar situation to you (and I trust the guy answering).

Anyway, hopefully one of those workarounds will do the trick for you.