It is my understanding the the Quote Notification list cannot be cleared.
Although I can't quite understand why one would want to clear the list, I suppose you could try sending a Comment to the admins, and explain why you feel there is a compelling reason to clear your list and see if they buy it.
BTW: This question has been asked on numerous occasions. One of the reason that you have not had any responses, aside from mine, is that you should have done a Search before opening the thread.