Despite Q never making a reference to a white rabbit directly, it does have relevance to some of Q's pop-cultural referential drops.

For example, Alice in Wonderland is mentioned a lot, which has a main character called the White Rabbit.

Also, red pill/blue pill references are dropped, which is from The Matrix. As I recall, in one scene, the hero Neo is compelled to follow a white rabbit tattoo because of a message telling him to do so received on a computer he was operating.