Do Quick Games Count for Missions in Naruto Arena?

I can't give a definite answer without knowing exactly which "Naruto Arena" game you're playing—there are multiple Naruto titles.

Mission requirements and the definition of "quick games" vary by game. I don't have access to specific game data for this title.

To Get an Accurate Answer

1. Specify the exact game name.
2. Explain what you mean by "quick games" (shorter matches, casual mode, etc.).

With those details, I can better advise how quick games affect mission progress.