I want to make a full page chatbot using Watson Conversation (like this: https://www.wmdmark.com/ ), is it possible to do it? If yes, how can I connect the Watson chatbot and my website?
That is possible and is a typical usecase. Take a look at the architecture diagram shown in this overview of Watson Conversation. The conversation service is invoked by your chatbot. There are samples you can work with for different programming languages, including Javascript. You could build a Node.js server app that serves webpages with a bot interface to users.