The Paladin's hidden appearances are some of the most complex ones to unlocked compared to other classed, but with some patience it is definitely doable.
Holy / The Silver Hand Hidden AppearanceThe Hidden Artifact Weapon Appearance for The Silver Hand is called The Watcher's Armament. It is unlocked by using the item Lost Edicts of the Watcher.
The drop location or the steps needed to get this item is still unknown. It dropped from demons in Azsuna in the beta version of Legion, but has been confirmed to have another source on the live servers.
Protection / Truthguard Hidden AppearanceThe Hidden Artifact Weapon Appearance for Truthguard is called Vindicator's Bulwark. It is unlocked by using the item Spark of the Fallen Exarch.
To get your hands on this you need to complete the Withered Army Training World Quest / scenario. This requires you do progress into the Suramar questline to a certain point, and after that you can spend Ancient Mana to unlock the scenario. The item itself drops from the final chest of the scenario, but it is not known in the drop rate increases by getting a better score. The drop rate itself is pretty low, do you most likely need to run the scenario a few times before you get it.
The best bet to get it is by farming 2,000 Ancient Mana and try to get the best score possible. Here is a short guide on how to farm Ancient Mana very fast.
Retribution / Ashbringer Hidden AppearanceThe Hidden Artifact Weapon Appearance for Ashbringer is called Corrupted Remembrance. It is unlocked by using the item Heart of Corruption. This makes Ashbringer sort of look like Corrupted Ashbringer, but it's still not identical, so owners of the original will still stand out.
To get the Heart of Corruption, you need to complete a pretty chain of events. The first step is to obtain two books: 1) Nat Pagle's Guide to Extreme Anglin', which is looted from dead players in the battleground Alterac Valley. This is also not bound to a character, so you can farm it on your alts or buy it on the action house. 2) A Thoroughly Read Copy of "Nat Pagle's Guide to Extreme Anglin'.". This book is looted from objects called A Dusty Tome, which spawns randomly around the Dire Maul instances. Do a run of all the wings, and you will most likely find it. If not, reset the dungeons and try again.
For the next step to be possible, you need to reach a certain Knowledge level one your artifact weapon. It's not currently known exactly what this knowledge level is, but it seems to be around rank 15. In order to get knowledge power, you need to complete work order from Sister Elda. Upon reaching the needed knowledge power, she will offer an option to reform the books, which you must do in order to progress the chain.
Once this is done, go to Blackwing Lair and kill Nefarian. Loot his head, and head to Dire Maul West. Here you need to talk with Prince Torheldrin. The next step is to go to Western Plaguelands and find Alexia Ironknife. She is in Chillwind Camp, so she's not too hard to find.
Head over to Hearthglen and look for the book named Grand Inquisitor Isillien's Journal. It should be in the townhall, the same building that has a clock tower. Coords are 42.36 18.73.
Now you need to go to the western river bank of Western Plaguelands and kill a slime. This slime is called Large Vile Slime, and will drop an item called Timolain's Phylactery. The slime has a pretty long respawn timer, and has a tag, so group up with the other paladins there.
The phylacery will spawn a new mob, Timolain the Accursed. Upon dying, he trows a Shard of Darkness into the water. This item is also needed, so you need to get out your fishing rod and be prepared to spend a lot of time fishing. Getting it might take several hundred attempts!
After getting the crystal, you need to head to Light's Hope Chapel and find Lord Maxwell Tyrosus. He will give you a quest called Walking in Shadows. This eventually leads you to Ebon Hold, which you can reach from the Dalaran Flight master once you have the quest.
It is not known if there are any further steps on the live servers, because no one has been able to get it this far, since it takes a while to get knowledge power. But in the beta this was the final step, and you get the Corrupted Remembrance skin! This post will be updated once more information has been obtained.