Keep list of websites in Notification Center I want to keep stackexchange site as a notification so that whenever I want to open the site I just need to scroll down from the top and click on link of the site. Is there any way to do that?
No, there is no way I am aware of of simply adding arbitrary links that do not expire once viewed.  You could create reminders or calendar links or any other sort of repeated notification which contain the URLs, but this is a most inelegant solution.

If quick access to the SE site is what you need, then you could put a link in your bookmarks menu or bar, or you could even place the sites in your reading list which means it's always in the top section after clicking bookmarks (assuming you don't use reading list for it's intended purposes.

If you want to open a URL without opening Safari first, the best option may be to go to the SE homepage, click the share icon and click add to Home Screen, and then either drop this into your bottom four home screen dock slots, or if you have more than one URL that you wish to use, then shove them all in a folder and put the folder in your dock.