If I want to run this sort of query in SQL Server, how can I do the same query from one server I am connected to to another?
I tried adding \"[ServerName1].\" before
The best way to do this would be to create a "linked server" between the two. You will need appropriate permissions to do this.
Then it's just a matter of accessing the databases using your linkedserver name.
Ex: [linkedserver].databasename.dbo.tablename
To create a linkedserver, go to server objects->right click on linked servers->click on 'new linked server'.