The word "god" may be either a common noun or a proper noun.

If that word is used as a name of a diety, or if it is essentially a title for the regnant diety of a culture, it takes a capital G.

If it is used as the name of a class of referent, such as "a god", or "their god" (in other words, a common noun), it takes a lower case G.

Kat was correct in her capitalization, although I can't think of a formal reason for the quotes.

Dave Oesterreich