Yes, it is possible, but you have to think hard about what it means for your application.
If an unhandled exception occurs, your application might be in an unexpected/undefined state , and it might make more sense to just crash the application - the alternative could be inconsistent or freezing UI, or even worse.
That being said, you don't state whether you are using WPF, WinForms, a Console app or ASP .NET, or something else. You need to tell us the technology you are using; but in general, you should be able to hookup a global exception handler, that just suppresses the exception.
For the background thread part, you can simply wrap your thread function in a try catch block.