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:
Where are the gods of Hamath, and of Arpad? where are the gods of Sepharvaim, Hena, and Ivah? have they delivered Samaria out of mine hand? Who are they among all the gods of the countries, that have delivered their country out of mine hand, that the LORD should deliver Jerusalem out of mine hand?
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.