First of all Big Bang is just a theory and might be proven wrong, or incomplete one day, so basically we don't know if this theory is 100% true.
I think quran supports this theory as I understand, see this verse:
Have those who disbelieved not considered that the heavens and the earth were a joined entity, and We separated them and made from water every living thing? Then will they not believe? [21:30]
Quoting from the Tafseer of Tabatabai (translation):
The repetition of separation of particles and atoms and what we see in the air and atmosphere guides us to a stage where the skies and earth were all one entity and Allah has separated them[...]
From the comments, it seems that some scholars have different explanations of the above verse so it might be not compatible after all.