Starting the QuestEdit
Speaking with Sextant, who is atop of Constellation Hill in the Heartlands, next to the monuments surrounding the Cloudopolis portal, will eventually result in him asking for a Looking Glass. As of Open Server Event 10, he ignores one already carried by the player until a second conversation.
Finding the itemEdit
At least two merchants sell looking glasses. Relic de Falco is in Cloudopolis and is relatively convenient to look for using a shortcut portal there. Alternatively, Silky Roads can be located at Midway Village, which is also in the Heartlands.