OK, maybe I wasn't completely clear in my example....the "TestTable" was in a different database (I referred to it as TestDB)

But to answer your question.....

This is (probably) not necessary
SELECT TestTable.Name from MySecondSQLServerDatabase.TestTable where MySecondSQLServerDatabase.TestTable.first_name = 'XYZ'

But this should work

SELECT TestTable.Name from MySecondSQLServerDatabase.TestTable where TestTable.first_name = 'XYZ'

But again...I *really* encourage you to use some type of proper alias. In all candor, it royally pisses me off when people don't do it.