I believe that God must have a different name elsewhere, such as in Psalm 19, and I don't just mean a "stone." How would I go about acquiring that information? I tried the Rock AS God.
You could check the Referred to As section on the Factbook page for "God". That has a list of all the verses for all the different ways that God is referred to. (8 for "my rock" and 4 for "Rock")