Do any people in the Bible self identify as atheists?
Or are they all theists, differing in which god they follow?
Apparently at least some atheists existed, since David had heard of them:
The fool says in his heart, “There is no God.”
King Sennacherip and his general Rabshakeh come closest to being self-proclaimed atheists in the Bible. Rabshakeh says in the name of the Gentile king to King Hezekiah:
Here Sennacherib denounces all supernatural intervention to save, not just of Yahweh, but of the dozens of Canaanite gods around Israel. He essentially is saying that the gods spoken of here, are not gods.
Besides this example, I must conclude that the answer to your question is no. There are no self-identified atheists in the Bible.
The NET Bible notes:
If you read through the text of these nearly identical Psalms, you will notice they concentrate almost entirely on moral foolishness rather than any sort of intellectual foolishness. The phrase "There is no God" seems more like hyperbole than a statement of fact. Nearly everyone in the pre-modern word would have been classified as theist in the broadest sense. A perhaps more representative verse would be:
My interactions with atheists (both online and off) suggest that this particular Psalm doesn't apply to everyone who says there is no God. Many people who claim the title "atheist" believe their actions have consequences. They do their best to raise their children lovingly and ethically. They support worthy causes and give of themselves to make the world a better place. They refrain from wicked deeds and mistreating others. In fact, this Psalm applies much better to certain people I know who claim to love God, but by their actions show they don't believe He exists.
Not exactly the answer you are looking for, but Paul clearly understands humanity to be functionally atheistic, if not in practice. From Romans 1:
In practice, to be a-theistic in the ancient world was to invite destruction upon the whole community. In Rome, in particular, for example, legally they didn't care which god you worshipped, as long as you worshipped one. I don't have the quote handy, but I remember David Bell in Many Mansions making the case that atheism was like "poor hygenie or a public health hazard," meaning that impiety would invite needless retribution from someone, so if everybody was worshipping what they thought, at the least the "true gods," would forgive them.