To follow the Task-based Asynchronous Pattern and usual naming conventions (methods should be verbs, not nouns), your Authentication method should be named something like AuthenticateAsync.

Also, if this was a real code, you shouldn't use Task.Run() on a synchronous method, instead the whole CheckIfNameExistsInDatabaseOnServer method should be asynchronous, since accessing the DB is IO-bound.

To follow the Task-based Asynchronous Pattern and usual naming conventions (methods should be verbs, not nouns), your Authentication method should be named something like AuthenticateAsync.

If this was a real code, you shouldn't use Task.Run() on a synchronous method, instead the whole CheckIfNameExistsInDatabaseOnServer method should be asynchronous, since accessing the DB is IO-bound.

The parentheses in your ternary expression are not necessary (also following Malachi's advice about newlines):

string outputString = await Authentication(usernameTextBox.Text)
    ? "Name exists" 
    : "Name does not exist";