I've watched some of quill18's gameplay on youtube and he was dealing with Egyptians and Incas that had incredible 700 gold income per turn. Meanwhile, 100-ish was what I was able to ever see as a player. Is AI cheating in some way (handicap?) or I just cannot see the way they do that?
Short answer: Yes, the AI cheats.
At any difficulty above Prince (He mentions he's on King near the start), the AI starts to get bonuses. For King, they get an extra starting defense unit, less unhappiness, faster growth, and all sorts of other benefits.
That having been said, from what I understand, none of the AI bonuses for higher difficulties actually give them more gold. I haven't watched the LP in question, and 700 gold does seem a bit high, but if the AI has reached that level, I would first suspect there is something going on as a result of the Vanilla Enhanced Mod he's running. I don't know the details, but I do know that the people behind the mod have rebalanced a lot of the details of the difficulty bonuses, so there might be some effect there which is helping them keep their income up.