Every athlete, coach, and parent has at some point requested better reaction time or quickness to be more successful in their sport. Quickness is best measured by reaction time, the duration between a stimulus and initiation of movement. Mel Siff, who wrote one of the most comprehensive sports science texts, Super Training, defined reaction time as 2 distinct phases, a latency period and a response period.
While the latency phase is mostly determined by genetics, the response time may be increased through training, particularly playing your sport. This sports specific training increases your skill of movement, increasing the ability to both anticipate and make decisions. So weightlifting will improve your repetition of these quick movements, reduce injury risk, and increase speed, but chances are that your quickness is already well established through years of practices/games.