I found out about Idge's armor upgrade from Kenmar's blog Journeys Through Granado Espada and the forum link he kindly provided here. My previous attempt to do this quest failed, when I brought a full team to the bonfire. When I succeeded, I was using her solo. So, maybe the quest won't trigger if you do not solo her.
- Click on any bonfire using a Lv.60+ normal Idge (not battle-smith), while wearing Lv.44 Idge's Breastplate. A bonfire can be found at Tetra Ruins Grand Corridor (B6), up the stairs. Another bonfire is in Jezebel Glen, near the colony.
- Unequip Idge's Breastplate and click on the bonfire again with Pure Aidanium x50, Pure Etretanium x50, Pure Ionium x50, and Pure Talt x50.
This piece of metal armor isn't superb, but it does not have ATK Speed penalty of most metal armors. It is a seldom completed side quest, I expect. I love secret, hidden quests~!