I saw this sentence being used on some websites "Flip open the first page of your book". I learnt that we should not use "Open the 2nd page of the book", instead we should use "Open the book to page 2 " or "Open the book at page 2". I am confused if "first page" has an exception or "Flip open" works for any page.
No, it isn't correct. You don't 'open a page', you open a book. Further, the process of locating a page is normally referred to as 'turning to' a page.
You could say:
Open the book and turn to page 10.
That is idiomatic, but a bit long - you can't turn to a page without first opening the book. So you could just say:
Turn to page 10 of your book.
Look at page 10 of your book.
It would be difficult to open a book at the exact page you want, so there would normally be a process of turning to a page. However, because your example is the first page, it would be more reasonable to say open it 'at' that page.