My current tenant doesn't want to move out after the landlord given 5 months notice to them. What should I do now?

I'm confused. You say "my current tenant" but then you refer to "the landlord given 5 months..."

If you are the landlord then you you do an eviction. You after to follow the legal eviction process for your area. Since you did not give us a location (ANY question on law needs to include a general locale as laws vary by area) or any details like is there a lease, have they been paying rent, what type of notice was given, we can't be more specific.

But if you are going to be a landlord BEFORE you start renting you need to familiarize yourself with the tenant landlord rules.