The main condition in order to start the quest of releasing the animals is that the territory the crystal is on must be yours. An exclamation mark will appear above the crystal when you can start the quest. If the territory was redeemed long before the quest was activated, the crystal might be placed in the Warehouse. Check your Warehouse to see if the crystal with an animal is there!
Keep in mind that only one quest can be active. Once you complete it, you can proceed to the next. The order of quests is as follows: Lucky the Leopard, Red Panda, Leopard, Fennec, Panther.