You need to check with the other plugin how to trigger the login modal to come up on clicking the like/unlike buttons. They must be having a function to be called to show the login pop up. You will have to track this in the javascript file that comes with our plugin and implement the same there.
You can also put html code there in the setting Login required message like Please <a href='#' class='popup-login'>Login</a> to vote. This class='popup-login' is as per the documentation of the other plugin you referred.