Here's one concept. There might be other ways of doing this. This is best for a unique hero unit, not something you're going to have multiples of because the triggers will get messy.
Ability 1: A dummy passive ability (based on Stormhammers or something). Really just a tooltip describing the effect for the player.
Ability 2: Active ability with a passive icon. This is the ability that actually does the thing. Give this the same icon, tooltip and button placement as Ability 1 so it looks like they're the same ability to the player. Give this the cooldown you want the effect to have. Remove any mana cost and hotkey.
(If you're doing the actual effect with triggers, this doesn't need to actually do anything except be active and have a cooldown.)
Trigger 1: Start of map, disable Ability 2.
Trigger 2: When <event happens>, enable Ability 2, disable ability 1, issue order to unit to cast Ability 2. Turn off this trigger.
Trigger 3: When Ability 2 is cast, start a countdown timer with duration equal to the cooldown. Any trigger-driven effects of the spell also happen here.
Trigger 4: When the countdown timer expires, turn on Trigger 2. Disable Ability 2 and enable Ability 1.
Edit to add - Here's how all this will look to the player:
While the ability is ready, the passive icon is visible. They can read the tooltip, and clicking it doesn't do anything (because it's a passive).
While the ability is on cooldown, the active icon is visible and the player can see the cooldown in progress. They can read the tooltip. Clicking it doesn't do anything (because it's on cooldown).