You can first start doing Elite quests around level 3-4, the Elite box must be checked in order to make the quest Elite, and that only works for the few quests that are Elite. I highly recommend bringing a partner along for this one, maybe get "Hold Me" out of the way as well. Elite is a pretty ridiculous increase in enemy strength and I have yet to see a reasonable quest to solo on Elite (granted I've only tried a couple of times).
The easiest method to accomplish this early and solo that I've seen is to do any of the Trade For quests (Delphi Limestone, Mosaic Tile, etc.) and just get to age 2 and amass a small caravan of... caravans using the Market and send them to the checkpoints 1 by 1. The only map where this might be a little difficult is the setup where you're in the corner surrounded by cliffs with a checkpoint right at the entrance to your little area if there is a large number of enemies there. Although, in general very simple.
It should be noted that you can save a quest and finish it at a higher level, and there are also several repeatable quests you get early on that can be easily done on Elite at higher levels. An example would be that I regularly do the Greek challenge "Keep Them Under Our Wings" on Elite as it can easily done by just setting up and walking away for the 10min duration, giving me time to take a break from the game or something while getting free xp and the upgraded Elite loot (which still kinda sucks at the higher levels but it requires no effort).