Let's say the 4 teams are A, B, C, and D, and let's assume the matches are chosen by random draw each week. On any given week, then, the probability that A will play one of B, C, or D is 3/19, because there are 19 other teams and they are all equally likely to be chosen. Let's say, for example, that A plays B. Then the probability that C will play D is 1/17 by similar reasoning. So the probability that the 4 teams will play each other on any given week is (3/19) * (1/17) = 3/323.
Since there are 38 weeks, the probability that the 4 teams will NOT play each other at all is , and the probability that they will play each other at least once in 38 weeks is