If I knew the context I could possibly give you a better answer. But it sounds like they could be talking about American football, so I will answer assuming this. To seize the championship means to be in a position to win the entire series. Down two means the other team is ahead by 2 points. In the fourth quarter means the game is timed (15 minutes per quarter) and this is taking place in the last 15 minutes of the game. I hope this is helpful.

'to seize' here means 'to win' The 'championship match' is the match that determines the winner of the competition or tournament. 'down 2' in this context means you are behind in the score by 2 points. ie. The other team is in front of you or leading by 2 points. Some games are broken into quarters. ie. If the entire game goes for 80 minutes, for example, each quarter is 1/4 of this period or 20 minutes. Sometimes, there's a pause/rest at the end of a quarter, sometimes they just swap sides of the sporting field. Here, in the 4th quarter means the game is almost over - they are in the 4th and last time period of the game. "San Antonio was in position to seize the championship down two in the fourth quarter." means San Antonio, although behind 2 points in the score in the final period, were in a good position to win.