I don’t think I ever say this word to human beings – I reserve it for when my cat is being particularly obstructive. It sounds nicer than ‘get a sodding move on’.
It has been suggested that the stick referred to is a type of walking stick, but it might be more useful to compare it to stick as a nautical term for a mast or yard. That’s the sort of stick referenced in the phrase up sticks, meaning ‘to prepare to move’, ‘to pack up and go’, or ‘to remove oneself’.
I especially like this early usage from 1867:
“This is a bad business, Bob: if that ‘ere doctor ain’t here pretty quick-sticks,..it’s all over with this chap.” Example Better than Precept, M. A. Mackarness.
Quicksticks has a modern usage as a type of hockey, played to introduce children to the sport. It has 4 players on each side and uses a large, light ball.