The bludgeon axon is one of three pieces that can be combined with the bludgeon spine and bludgeon claw to make an abyssal bludgeon by speaking to The Overseer, who combines the three parts for the player. If a player has already had the Overseer construct one, they will be required to use the parts on the Overseer's book instead.
Players have a 62/128 chance of receiving one via using an unsired on the Font of Consumption behind the Overseer. If a player already has a bludgeon piece, they will not receive duplicate pieces. For example, if a player has obtained a bludgeon axon and their unsired rolls for a bludgeon piece, they are guaranteed to get either a bludgeon claw or spine.
For an exhaustive list of all known sources for this item, see here