Changing background color of HTML elements based on user input
The first thing that jumps out at me is the tight coupling it has with the current application in which it is being used.  Why hard code all those CSS selectors?  This class really cries out for configuration.

You might see a pattern like this used to achieve decoupling of class from config.

function ArticleQuiz($el, $config){

   this.config = {};
   ...
   init() {
       $.extend(this.config, this.defaultConfig, $config || {});
       ...
   }
   this.init();
}

ArticleQuiz.defaultConfig = {
    answerDataLabel = 'answer',
    answerContainerSelector = '.answer-container',
    ...
}




You should consider taking the approach of using meaningful class names to apply multiple CSS styles at once.  So, rather than using specific semantic color and font class names, define a CSS class that encapsulates the entire state of being a correct/wrong answer.  So taking this approach along with the suggestion to store configuration something might allow you to change this:

        $(this).addClass('background-' + ((THIS.answer === false) ? 'green' : 'red')).addClass('white-font');
        THIS.result.addClass('font-' + ((THIS.answer === false) ? 'green' : 'red'));
        answer =  THIS.answer ? THIS.incorrectAnswer.show() : THIS.correctAnswer.show();


Into something like:

       var classToApply = THIS.config.correctAnswerClass;
       if(THIS.answer === true) {
           classToApply = THIS.config.incorrectAnswerClass;         
       }
       $(this).addClass(classToApply).show();




Your QuizLogic function would not seem to have reason to be in the static (prototype) context, as you are using it as a constructor for your class. I would think this function should be within class function scope instead on prototype and should be call something like init() or similar common name for constructors in javascript.  Note how I present the init() function in my first code example above.



Consider first populating your instance properties in the constructor rather than in the main class code.  This gives you the ability to do things such as validate that the selectors do not return empty collections and similar activities needed to put the object in the proper state for use. If these validations fail, you could throw an error to prevent the object from being used in an improper state.