Sternfathoms Pet Journal is a new way to introduce a pet battle quest in Legion.
The item as you can see is a toy located in a treasure chest in Aszuna. Below map shows you the location, and to make it even easier, Cymre kindly created a video guide how to find it.
Using the journal in Aszuna summons a Shipwrecked Captive who offers a weekly pet battle quest that yields 4 Pet Charms when completed.
The fight itself is not very hard and can be used to level pets. Currently in the beta, as long as you do not hand in the quest, the fight can be repeated endlessly and is therefore a very good source for pet experience.
This is very likely to be fixed at some point!