I'm trying to figure out what the correct feminine forms of "god king" and "god emperor" are. Do I just replace "king" and "emperor" with their feminine equivalents (i.e. "god queen" and "god empress"), or should I also replace "god" with the feminine version (i.e. "goddess queen" and "goddess empress")?
If you have questions like this, I suggest you use websites such as the Google Books Ngram Viewer. This allows you to search all published works available to google for words and phrases.
As for an answer to your question, I did the search for you and I found that goddess queen and god queen have minimal but some usage. Goddess empress and god empress have none. God queen appears to have had more usage in the past, but modern usage has dropped to a similar level as goddess queen.
I also attempted to look for divine queen and divine empress, but they weren't any more popular than their god(dess) counterparts.
To make it simple, there is no officially "correct" form. It depends on each individual ruler. There is even precedent for women to adopt male titles as rulers, such as the female pharaoh's of Egypt.
On a personal level, I immediately gravitated to god queen myself. While goddess is exclusively feminine, there is plenty of precedent for interpreting "god" as a gender neutral term, especially in monotheistic religions. For example, in Christianity, even though God is often discussed using masculine pronouns, God actually has no gender. As a side note, within the Bible, God is occasionally referenced using feminine pronouns as well.
This is further complicated by the specific situation you are talking about. Is the ruler a reincarnation of a particular god/goddess? Are they descendant from a divinity? Are they the literal embodiment of a deity (possession)? All of these factors can change how that individual ruler identifies.
I recommend if you're translating a foreign title, just stick to a literal interpretation of the name. If the term is gender neutral, use deity instead of god/goddess. If you are inventing a title for a fictional character, then consider what traits you want to portray and what the motivation is of that character.