I use fb:comments on my website and I also have a Facebook Page. Is it possible that whenever a user posts a comment on a webpage in my site, the same comment gets posted on the Facebook's Page wall?
No it is not possible, you cannot control where a user decides to post his comment. This will negate comment ownership, the user did not authorize the website to post on the page wall only in the comments plugin.
You can however maybe include a Facebook tab with the fb:comments shown. If the fb:comments span more than one blog post then you may need to create your own app within the tab to handle this.