That is why I asked if it is a full-on quarterstaff or just a walking stick. A quarterstaff is iron-shot (or tipped or something). A long walking stick (or light staff) counts as a club. Which still has Encumbrance.
How weak is the character that you are so worried? If you are playing a male Flambeau with a negative Strength and no martial skills, then I shake my head and cast doubt on how long your character will live.
Me. I am playing a female Flambeau with negative strength and a small size. But she is a girl, and carries no weapons in the first place. Her father always discouraged her from the manly art of combat. Still, I might have her take up the Short Sword one day, after the example of Rulia of Flambeau.