"The fool has said in his heart, “There is no God.”" (14:1). Is the psalm talking about someone who argues that God doesn't exist?

The psalm gives only his words, where he says them, and what goes with them. He says them "in his heart" (14:1), and the verse goes straight on to deeds: "They are corrupt. They have done abominable deeds" (14:1). Later the psalm says the workers of iniquity "don’t call on the LORD" (14:4). Brown-Driver-Briggs gives naval, the word for fool, as "foolish, senseless, esp. of the man who has no perception of ethical and religious claims", and Strong's as "stupid; wicked (especially impious)" (tap fool). Elsewhere in the Psalms the word is used of those who mock God: "Remember how the foolish man mocks you all day" (Psalm 74:22). Psalm 10 lets the wicked speak in his heart too, and there he names God: "God has forgotten. He hides his face. He will never see it" (Psalm 10:11); the chapter on Psalm 10 looks at those words. Psalm 14 doesn't say more about what the fool believes than his words, and these notes don't either.