Patch 2.4.1 added lots of non-combat pets, including the Liv Moore pet. This pet is based on the character with the same name in the iZombie television series, and is dropped by a unique mob named Ravi Liliywhite.
Ravi Liliwhite is found in Weeping Hollow in Act 1, and spawns in Adventure Mode. He might also spawn in Quest Mode, but that has not been confirmed. He can spawn anywhere in the Weeping Hollow map, so if you are farming for the pet, search all of it before you create a new game.
The mob Ravi Liliwhite has a really low (about 1 %) chance of spawning, so it is extremely rare. This means that you will need to create new games and search Weeping Hollow a lot of times before he will spawn. Once he has spawned, he can be killed very easily, and will drop Liv Moore 100 % of the time. So the spawn is rare, but the pet is guaranteed to drop.
The best way to farm for the Liv Moore pet is to have a fast-moving class that can rush the whole map in around one minute. Search all the mobs there, and restart a new game if Ravi Liliwhite has not spawned. There really isn't any more tricks to it, so good luck with the farming!
The same patch also added 19 more pets which all drop from the Menagerist goblin. Read how to farm the Menagerist Goblin for pets here.