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The Lucky Looter perk does not change the quality of quest rewards. There is a rare chance of getting a high quality item as a tier 4 quest reward, but the Lucky Looter has no effect on this.
Currently for a tier 4 quest reward, you would get quality 2 tier leather/iron armors, iron weapons, and iron tools, but there is a rare chance that you would get high quality military/steel armors, steel tools, and steel weapons with or without points in Lucky Looter.
This is how the quest rewards are now, but it's still wip, so it may change in a future version of the game.