Yes, you need a champion immediately adds turn meter to the team via a skill. As EGDNIT mentions above, an example would be High Khatun, and there's quite a lot of other ones too
Check out the following list to see if they have any: https://ayumilove.net/raid-shadow-legends-increase-turn-meter-skill-champions/
If you are running a team of 4 champions then a single cast of it will count as +4 to the quest, so after 13 uses of the skill they should get the quest, though it doesn't need to be done in the same arena battle. Going from memory they will need to win each of the arena battles that they use the skill in too