Unfortunately the "Tournament" in Luca during the main story does not count towards this. After finishing up in Luca you will gain access to play Blitzball at any save sphere in the game. After playing some regular season games you will notice "Tournaments" will become available. They usually consist of winning three consecutive games.
If your wanting to get this as early as possible. I recommend recruiting (Wedge) from Luca (Replacing Datto). He has a very high starting shoot stat and combined with Tidus's (Jecht and Sphere) shots you can put up big numbers on low level teams.
I also recommend replacing (Keepa) with (Jumal) located in Luca Plaza (Near the tavern). He is a excellent starting goalie and makes it very difficult for teams to score on you starting out.
After making these key acquisitions you should be in good shape to win a tournament. Just be wary of the Al Bhed Psyches as they have the best Goalie in the game (Nimrook) and can be very difficult to score on for inexperienced players.