Pigeon Jesus refers to Amynhotep's OC Akra Shizuo. Due to his appearance and his massive wings, he was soon called Pigeon Jesus to everyone and acts as substitute for the lack of religious figures in the group (At least in the crack way).
Often, the member Amy gives around blesses under the name of pigeon jesus when new members arrive.
Amy: But only if you are truly worthy of Pigeon Jesus's blessings.
Rumors say that if you tell your prayers to Akra, they might come true. Maybe.