They are not directly related.

The relationship today comes from the recent forms of protest held at Tian'anmen Square. Tian'anmen is a frequent site of protest by people from a wide range of backgrounds, even if there is no actual connection to the happenings of 1989. A large number of protests target the anniversary of the June 4th Incident, and has frequently been one such attempt.

Falun Gong, called Falun Dafa by its practitioners, started out in the early 1990s as a variation on qigong, itself not to dissimilar to tai qi but with a strong sense of healing attached to it. This wasn't until a few years after the Democracy Movement. It was only in 1999 that FG came into strong opposition with the Central Government. The organisation demanded official sanction, as given to mainstream religions in China, but the Party did not agree to it and it was from this point that they started actively working against FG.

Things have gone down hill between the two sides from there, but that is of course outside the 20 year rule.

Despite it being too recent to talk about here, there are plenty of comprehensive articles written about the history of the movement, including Wikipedia, which under normal circumstances I wouldn't direct people to in an AH answer, but again, there's that 20 year rule. You should be able to find more recent events explained pretty clearly there.

Hope that helps, and sorry that I can't really get much later in time than 1999.