Tz'arkan's Whisper Quests have a minimum cooldown and a maximum cooldown (turns).
Default values are minimum: 10 and maximum 20
However neither of these values are what they seem to be.
The best way to explain the mechanic is via a table and then an explanation:

In a new game after your 2nd turn initial quest your odds to get a new quest look like this:


HOWEVER:
The stats don't appear to be in our favor for getting a new quest, but what I'm about to reveal makes it much worse.
If you get lucky and happen to roll the 5% you are not guaranteed a new quest. Why, you may ask? Well, because each mission type has its own internal cooldown.
The cooldown of each mission is the same value as the mission timer you see displayed in-game. However the caveat is this timer does not decrease each turn as you would expect it to. It decreases only when you win a roll for a new quest.
When this roll is won you are either:
A) rewarded a new mission and each current mission timer is decremented by 1
B) not rewarded a new mission (all are on cooldown) and each timer is decremented by 1


I changed the quest cooldown timer to 5 (from 10)
I changed the maximum cooldown to 5 (from 20)
I added logic to decrement the internal mission cooldowns each turn. (which I believe may have been the original intent of CA, because the UI does not support two missions of the same type)

With this mod your chance to get a new quest after turn 2 will look like this:


Since the new quests will actually be coming in between 6 and 10 turns after your last quest it is possible that the internal quest cooldowns prevent you from getting new quests (raze missions cooldown can be 25 turns, kill X missions are always 20 turns) However as I stated above this was probably the intent of the internal cooldown timer added by CA because the logic that governs the UI was not intended for there to be more than one mission of any single type active at a time.

Tz'arkan's Whisper Quests have a minimum cooldown and a maximum cooldown (turns).
Default values are minimum: 10 and maximum: 20
However it is the way the game calculates your chance to get a new quest between 10 and 20 that makes it unlikely for you to get one on an earlier turn.

In a new game after your 2nd turn initial quest your odds would look like this:


I simply changed the minimum cooldown to 6 and maximum cooldown to 12.
I also increased your chance to get a quest by 10% rather than 5%.

With this mod your chance to get a new quest after turn 2 will look like this: