|LARP staff. Best I could find|
Looking at my own rattan practice staff, which is your traditional 6' long, 1.13" diameter lightweight stick, you're looking at about 1.35 lbs. That's a density of about a quarter-pound per foot, or about 0.52 g/cm3.
An oak staff of similar thickness would be about half again as heavy (0.75 g/cm3), and probably weigh about 900g, or about two pounds. Still two pounds short. Lignum vitae, the densest wood, might come in at about 3.3 lbs.
Increasing the diameter to a full 1.5" (quite stout) has the bigger impact, of course. Rattan would be 2.4 lbs, oak 3.4 lbs, and finally, hickory or LV would be 3.8 and 5.5 lbs, respectively. You could get certain kinds of teak to be betweeen 4-5.5lbs if you looked for pretty exotic woods.
Anyway, so your typical six-foot combat staff is probably 2-3.5 lbs of pure wood, which leaves 0.5 - 2 lbs of . . . what?
An alternate answer is that the quarterstaff in basic is not a six-foot "bo" type staff - which from some limited research of European weapons, it was not. More like seven to nine feet. George silver distinguished between the quarterstaff at 7-8 feet and the long staff . . . at 12 feet.
Now, a 7.5' long 1.44" staff, gripped at the back and the first quarter (one of the possible etymologies for the term quarter-staff) would extend about 67" past the top hand. Conveniently about two yards.
Anyway, point is, the quarter staff in Basic is either missing mass when made from common woods, or it's not the six-foot staff.
The other option (and it's likely a both/and rather than either-or) for that missing mass is of course, steel shodding on the end caps. A layer of steel 0.1cm in thickness about 9cm wide and 30 cm long would be about 211g or about a half pound. At the low end, you get two shodding ends, about 6" in length at each end, or the heavy end is a foot of metal, 2mm thick and about a foot long. That's a decent striking surface in either case. One could also use 1mm of steel two feet long, or even two feet of unshod wood at the one end of the staff, and the metal cladding being the top two-thirds of the stick.
In any case, there's all sorts of possibilities for what a GURPS quarterstaff could be.