You must have +20,000 faction with the Court of the Coin to unlock this quest. Once +50,000 faction has been achieved, however, this quest will no longer be available.
While any monkey will suffice, there is a plethora around the grassy courtyard of The Golden Scepter.
Capturing monkeys is more difficult than it seems. In order to successfully capture a monkey, do as follows:
Kill off any surrounding monkeys near the one you want to trap.
If there are too many surrounding monkeys, they will destroy the trap causing the trapped monkey to turn into an irate monkey. Furthermore, you can only turn in one monkey at a time.
Right-click the trap in your inventory to place it on the ground while near a monkey.
The monkey will automatically go inside, changing 'an empty monkey trap' into an occupied monkey trap.
Pick up the occupied monkey trap by left-clicking and bring it back to Abil. He will give you a new empty monkey trap.
Even though the trap will change it's name from 'empty' to 'occupied', if you pick it up too quickly, the trap will remain empty in your inventory. To prevent this from happening, you can wait 30 seconds or lure additionally monkeys into the already occupied trap. The mechanism is finicky.