Quechua's are really amazing rushers, my personal choice (over dog warriors, axemen, praetorian, immortals, war chars etc). The problem is they are really only effective until axemen show up, and they don't fair as well against hill archers / garrison. They have a base strength of 4.4 against any archer unit (combat 1, 100% modifier).
You're still going to have a hell of a time against protective civs, figure it will take double the assault force to take a city. Some civs aren't worth rushing like sitting bull, hammurabi's bowman. Most civs won't get axeman for a long time, even if they have copper near the capital. This will vary based on the difficulty you play, but you're essentially trying to beat the AI to the 3rd/4th archer, culture reaching 40/60% in capital, and any garrison promotions from fending off barbs.
So how do you pull it off on emperor+? First of all, the slower the speed, the easier the rush since your Quechuas will outdate slower. On normal speed its still quite viable to take out 1 or two immediate neighbors, but cross more than 20 tiles and you're going to need a solid army of 10+ since most civs will have 3 cities up packed with archers. I generally play on epic speed and am able to take out the most immediately threatening (or rewarding in capture) neighbor with 6 Quechuas. Aside from a barracks, these 6 are always my first build from the capital. If I have a fair amount of food/animals or trees to chop, I may go for an immediate worker. But in trial and error I've found that its best to just steal an enemy capital and its workers ASAP by using your cap as a quechua factory.
A lot of times you can pick up a holy city, stonehenge, great wall this way. If you're real lucky you'll be able to take out a 2nd civ, piggybacking on the xp your surviving units got from the first. By now they'll have cover and city raider I/II and can really lay down punishment on non-hill cities. You'll have to raze some cities and the distance will hurt you in the short term, but HC is financial and you'll recover quickly. You'll have more healthy/happy resources early on, a lot of breathing room to settle between your cap and the newly captured cities. The best thing about quechua rush is that you'll know within about 5 minutes of making the game how its going to pan out for you. Plus its fun to remove Monty and Mansu from the planet before abraham spoke to god on the mountainside.