Synthesize Gems is a quest unlocked as soon as the player has completed Mysterious Gems.
The objective is to get used to the mechanics of item synthesis. It is completed as soon as the player opens the Blacksmith window and synthesizes any item (the quest title/description suggests synthesizing a gem).
Nontheless every synthesized item completes the quest. Therefore it is also possible to synthesize a Luck Stone or any other doable recipe.
Click on the [Blacksmith] -> Select [Synthesize] -> Synthesize anything once. You may synthesize gems to strengthen them. Synthesized gems will give better attributes to your items.