In order to unlock this quest, you need to spend at least 500 gems of any type on regular gem trades with the collector.
Hatching any Zygarde egg will count towards your Shiny Chain!
You can shiny hunt them like any other regular shiny.
After you unlocked the Zygarde Quest by trading any 500 gems with the collector, you will be able to merge the first set of gems to get a Zygarde Core egg. After merging those gems three-times, you unlock the next forme of Zygarde. For each forme you will need a larger gem variation. To get the full list of gems for each Zygarde forme, have a look at the spoiler below: