Psalm 14:1 —

The fool says in his heart, “There is no God.”
They are corrupt, they do abominable deeds;
there is none who does good.

James Luther Mays, Psalms, Interpretation, a Bible Commentary for Teaching and Preaching (1994), pages 81–82:

The sentence seems to say that atheists are silly and atheism is frivolous. We know that in our culture that is not so. The denial of the existence of God is made by serious and honest people. In the society that this psalm describes, however, nabal does not mean things like dumb, inept, silly, clown, buffoon. Rather, the term designates a person who decides and acts on the basis of the wrong assumption. . . . A nabal is a person who, whether shrewd or powerful, makes a mistake about reality.

The “foolishness” with which the psalm is concerned is to say in one’s heart there is no God. That may sound as though the psalmist lived in a secular society and endured atheists who denied the existence of God. But the rest of the psalm makes it clear that the problem is not a reasoned intellectual argument against the existence of God but conduct based on the private assumption that human beings are not held accountable by God.

The psalmist reasons from the way people act to the way they think.

If people enact life in corrupt and perverse ways (vv. 1–3), do not pray to God in their need but live by preying on others (v. 4), then they are denying the reality of the LORD, the God of exodus and the covenant and the prophets. . . . It does not address, therefore, the phenomenon of modern atheism directly. But the “atheism” it does uncover is more dangerous, insidious, and general because it is a reasoning that can be found, as the prophets and Jesus insisted, in the hearts of the religious as well as the secular.

The psalm is not concerned with the question of whether people accept the existence of a supreme being. It is concerned with whether people acknowledge the reality of the LORD, the God of Israel, by calling on the LORD in need and seeking the LORD in the decisions of life.

Indeed, this “no God” as “no culpability” line of thinking is more flagrantly connected in Psalm 10:4,13.