This is an English language happenstance that often gets remarked upon - "dog" and "God" being perfect reverses of one another. It seems like just a simple coincidence. . . until you start looking a bit into folklore.
In the Native American traditions, dogs figure largely. My favorite is a creation story from the Kato Indians of California. They tell of the Great Traveler, who was going around the world creating things. He took a dog along for company. Note that he didn't create the dog. . . the dog coexisted with the Great Traveler. It was simply unthinkable that there shouldn't be a dog. It was probably a dog similar to those the Kato kept - small, with a pointed face and short hair, looking much like a coyote.
The Jicarilla Apache do believe that their creator, Black Hactcin. made the dog. It was the first creature he made, taking some yellow from the sunset to make a patch above one eye and some white from the morning sky to put a white stocking on each paw. These marks signified that the dog would protect people forever, both morning and night. Then Black Hactcin made man as a companion for the dog. When man laughted, the dog was very happy and ran around in circles.
The Yurok believe that people descended from dogs.
A Seneca chief is quoted as saying "It is most true that whenever a person loves a dog, he derives great power from it. But if you do not love a dog, he has the power to injure you by his orenda" (the holy, mysterious, unknowable forces of the universe). And the Hupa, Yurok, and Papago all believe that the dog has the power of speech, but chooses not to answer because to do so would portend a great calamity. The Papago, in fact, believe we would all turn to stone.