According to the Minecraft Wiki regarding Tridents, they should be able to be enchanted with Unbreaking. This is not the case when using an Anvil and an Enchanted Book.

The attached image shows me trying to add Unbreaking III to a Trident and it not allowing me to enchant it. Please note that I have tried this with my other encnated Trident and also a unenchanted Trident and have had the same results, therefor the enchantments are not causing the issue.