Any quest that shows a pair of dice/marked as random for the reward may turn into diamonds. There are some other rules (i.e. if it is labeled as a recurring quest, you only get the diamonds once per era per quest).
There are also some diamond rewards from taking control of specific provinces on the continent map.
Occasionally upon completing a battle in the PvP arena (win OR lose), you can get diamonds.
There are also diamonds from other places (such as great building effects, rewards from certain things in the Arctic Harbor, Oceanic Terminal, etc.).
Basically, don't worry too much, you'll get them as you play provided you don't blow them on things you don't need (i.e. don't ever buy something that costs diamonds from the construction menu). I've only been playing for two years and I have nearly 50k diamonds that slowly but steadily increase.
The game tends to be slow but if you believe you are going particularly slow you should ask around for a decent training guild as that will help you a lot. There are also some guides in the forum that would help you get a city off the ground faster.
Don't discount story quests and side quests. If your CF is robust, those diamonds you earn are boosted. You can also earn diamonds doing recurring quests, 1 time only for each quest you do in each era.
This game is designed to require patience. It is not one that you can rush through to get to the end as it has no end as such. I played for about 7 years before I bought any diamonds at all to play the game. The best way to be able to progress at a decent rate is to get yourself into a good active guild and be active in the guild. That way you will get all the help you need to grow your city without spending any cash.