No, the quality of a weapon only affect the physical damage part of that weapon (local). The attribute you are describing (skill gems on this item are supported by lvl 30 echo) is the implicit mod of Pledge of Hands (unique). Some items may have implicit mods, but cannot be improved in any way.
As a side note, you can change or add (to those items that don't have) an implicit mod with 'Vaal Orbs', but are very dangerous to use because they can change your current implicit mod (if there is one, so only use it on items with non-interesting implicit), change one/multiple of your socket colors to white, or reroll your item and change it completely. That mean that, if used with uniques, a Vaal Orb can break your item and convert it into a rare item (yellow). Don't use it with your Pledge ;).