Short answer: Nope!
While it might seem like a quick fix, most wood glues or glue in general contain chemicals and toxins that are dangerous if chewed or ingested by your feathered friend. Even once dried, some glues can release harmful fumes or flake off into tiny pieces birds might swallow.
š¦ Birds = Chewers. And chewers + glue = potential health hazards.
It's best to avoid any glue being used on toys.
Instead of glue, stick to natural, bird-safe fasteners like untreated natural cotton rope, paper rope, stainless steel hardware, natural hemp rope, or sisal.