[Recap - List of Qualities:
White = Merchant = 0 Add att(s)
Green = Common = 1 Add att(s)
Blue = Uncommon = 2 Add att(s)
Yellow = Epic = 3 Add att(s)
Violet = Legendary = 4 Add att(s)
Sienna (light brown-ish) = Artifact = 5 Add att(s)]
There is a tab called "Improve Equipment" in the Upgrade interface but that only resets and changes additional attributes. The second option in that tab turns Legendary items in Artifact items with a certain item.
Apart from that, i'm not sure whether there is any way to upgrade an items Quality any other way
You get runes from Danae's "Progression Trials". Just keep leveling up your character to unlock more of her quests and you will get enough runes.
Note that the Armor type items (Pauldrons, Breastplates, Boots) generally have more available runes and they are unlocked quicker than the attack type items (Weapons, Gloves)