How can I assert this kind of method works correctly?

You can create a test for the 2nd case, and it is OK to not have an assert in the test. It is because the test will pass when no exception is thrown, and it will fail when an exception is thrown. Sample code attached below.

For details please check Typemock resource What if I’m just testing that no exception is being thrown? section in https://www.typemock.com/always-assert-something/.

Btw, it seems NUnit supports Assert.DoesNotThrow which makes the test code more readable, but not sure if you could use NUnit. Ref: https://stackoverflow.com/a/23062325

Please let me know if you need further help or clarification. Good luck!

[TestMethod]
public void Validate_Called_DoesNotThrowException()
{
    // Arrange
    string userName = string.Empty;
    string password = string.Empty;

    // Act
    MyCode.Validate(userName, password)

    // No assert because we expect no exception is being thrown
}