In other words, why do muslims and christians tend to resort to the qur'an and bible when attempting to show the existence of God?

I think they do that because it is easier to talk about God from pages of the book than from looking at the world around them. The deity is more compatible with the inanimate and static literature.

Anyway, the qur'an or bible give reasons why God should exist, which the faithful agree with. They also agree that God must exist outside any religious book, otherwise he is the tooth fairy. So if God exists outside the holy book, the reasons for his existence must also exist outside the holy book, right? And since holy books appeal to the reason of believers, the holy books endorse reasoning too.

Why then do believers, in showing that God exists, require the suspension of reasoning, in favour of faith? Why do they argue from within the book, rather than from without, which is where the supposed God is?

I would have thought that the holy books would only be used when trying to show a person why one faith is better than another, since almost everybody has their own book. It is the other way round instead.

Also, in cases where they come across atheists and the like, since God exists outside any book, would it not be a start for them to show that God exists first before asking the atheist to put his faith in any particular book?

To me, always falling back on the bible or qur'an means you can't defend God's existence without that book, while the reverse should be the case.