What drives the Garlean war machine is, like Zenos says, mostly fear. They fear the consequences of religion, though it's important to note that their own actions often drive the greatest extreme (primal summons). One can argue that it's because they are aware that primals are one of the few things that can get in the way of their conquest, or that it's out of concern for the safety of the world... the truth likely lies somewhere in-between, but either way what they're doing is not helping matters. Regardless, fear is fear.

The Garleans have no known religion because they cannot use magic; somewhere (Encyclopedia Eorzea, I think - don't have one of my own) it was said that religion is a consequence of magic, so the magic-less Garleans being atheists makes in-universe sense. Lucia also suggests that the Garleans' atheism is a product of selfishness, as she genuinely adopted the Halonic faith after falling in love with Aymeric and going native in Ishgard. (Her exact line was something like, "You may think it strange for a daughter of Garlemald to be praying. But when I discovered something more precious to me than my own life, praying felt like a natural thing to do.")