"They do not believe in any god."
Translation:Ils ne croient en aucun dieu.
"Pas" is not the only possible complement to "ne". Here are some others:
ne...rien — not anything, nothing
ne...personne — not anyone, no one
ne...plus — not anymore, no more
ne...jamais — never
ne...aucun(e) — not any, no
ne...que — only
'Aucun' must go in front of the noun it describes (see Tips & Notes for the Adjectives 3 unit). This is because unlike, for example, 'pas' and 'jamais', which are adverbs, 'aucun' is an adjective (and a determiner).
dieu – god
aucun dieu – not a single god
Ils ne croient en aucun dieu. – They do not believe in any god(s).
I believe in God. – I believe that God exists.
I believe God. – I believe what God says.
That's the difference in English, and the same difference holds in French. The French even capitalize the word "Dieu", like we do in English with "God", when they're talking about the one god of the monotheistic religions.