While the exact path of the word's evolution is not entirely clear, it's likely that the sense of "to shake" evolved into a sense of "to quarrel" or "to argue" as the verb came to be used metaphorically to describe the agitated movements and sounds of people engaged in a heated dispute.
The first recorded use of the word "squabble" in English dates back to the early 15th century, and its meaning has remained relatively consistent ever since.