The short staff [ie. quarterstaff] or half pike, forest bill, partisan, or glaive, or such like weapons of perfect length, have the advantage against the battle axe, the halberd, the black bill, the two handed sword, the sword and target, and are too hard for two swords and daggers, or two rapier and poniards with gauntlets, and for the long staff and morris pike.

-- George Silver, Paradoxes of Defence

Note that he equates the staff to various polearms of similar size that can be wielded using similar techniques. For the purposes of combat these weapons were interchangeable, so soldiers in warfare would generally be using staves with metal heads that made them into bills, glaives, and so on. But quarterstaff was the core technique that they were all based on.

The deadliness of staff combat was perhaps most convincingly demonstrated by Richard Peeke in 1625, who claimed to his Spanish captors that (like Silver said) an Englishman with a quarterstaff alone was superior to Spanish swordsmen. Peeke went further than Silver: he boasted that he could best any number of swordsmen at once, using only a staff with no head. The Spaniards took him up on the boast, removed the head of a halberd, and had him fight 3 sword-and-dagger-armed Spaniards simultaneously. Peeke defeated all 3, killing one of them in the process. And that was without the weapon head. In battle with proper warheads, these pole weapons were extremely dangerous.

Quarterstaffs are present in a wide variety of medieval fechtbucher, and continue to have a presence in later periods, as well. It's difficult to give a general answer, as many of these manuals were written in different periods with different goals in mind, but very generally, it seems that quarterstaffs were used often as training weapons for other polearms, like pikes and halberds.

George Silver, as mentioned by u/BlueStraggler, is one example, but he's also somewhat problematic, as much of his martial advice seems to be aimed toward discrediting non-English martial traditions.

German manuals like Joachim Meyer and Paulus Hector Mair, both of whom published their manuscripts in the mid 16th century, include substantial sections on polearms, and Meyer himself states, at the beginning of Part 5 of his manuscript:

"...I will first cover the quarterstaff as a basis of all long weapons..."

Meyer's manuscript was an attempt to teach an interconnected system of fencing, which included the use of the longsword (a two-handed weapon of varying lengths that might be more popularly known as a "bastard sword"), the dusack (a short saber), the rappier, dagger, and staff weapons. All of these weapons in his manual shared postures, cuts, thrusts, and more importantly, the systemic concepts that Meyer had synthesized from earlier manuals. In his dusack section, for instance, he states that the dusack is:

"the basis of all weapons that are used with one hand."

Which is almost exactly what he says about the staff. Now, we know that the dusack was absolutely a battlefield weapon, but even here it's being taught as one element in Meyer's complete system.

In that respect, Meyer is teaching his weapons both as a means of personal self-defense, their use in duelling, their use in "fechtschuler" or popular festivals in which skilled martial artists would demonstrate their skill and compete in non-lethal tournaments, as well as, arguably, their use on the battlefield.

A couple of clarifications and expansions, here. Meyer is sometimes regarded as an outlier from the standard German martial tradition that started somewhere in the 14th century by Johannes Liechtenauer. His zettel, which is sometimes translated as "the recitation", was a poetic explanation of swordsmanship, manly virtue, and chivalric behavior. Nearly every German martial manuscript follows this tradition, and glosses many of Liechtenauer's verses, attempting to explain or expand upon them. There are dozens of these manuscripts.

Meyer, who wrote in the mid 16th century, is often argued as being removed from this tradition, although his manuscript quotes directly from Liechtenauer's zettel. The reson being that many consider Meyer's system of fencing as being something useful only for displays at fechtschules and for other non-lethal fencing (one of the main reasons being that Meyer states in his longsword section that "thrusting is abolished amongst us Germans" but happily teaches thrusting as a means to start attacks with the longsword, and he has no problem teaching the thrust as core to the dusack, rappier, and polearms).

B. Ann Tlusty, in her The Martial Ethic in Early Modern Germany however, makes a concrete case that swords and swordsmanship were far from useless in mid 16th century Germany, and Meyer's prohibition of the thrust may have had more to do with demonstrating your desire to defend yourself, as opposed to just murdering someone in the street. Punishments for murders were much more severe than for manslaughter, and since thrusting was far more dangerous to your opponent than a controlled cut, use of the thrust might argue that your intentions were murderous.

Another theory is that Meyer's longsword, far from the sporting weapon that many modern Historical European Martial Artists regard it, was actually a way to synthesize the Liechtenauer tradition with the battlefield zweihander, which was a weapon used to break up pike formations. Meyer's system uses many cuts and "handworks" that attempt to keep the weapon moving, which would be of great use to the long zweihander.

If we agree with the latter theory, then his use of the quarterstaff as a training weapon for a pike, halberd, or other polearm make it very useful for battlefield use. It was unlikely, though, that the staff itself would have been used on the battlefield in any significant degree.

Outside of the German tradition, the Italian Fiore dei'Liberi, thought of as a likely contemporary with Johannes Liechtenauer, taught a substantial section on the spear that utilized the kind of percussive cuts that would certainly be of use to a quarterstaff.

tl;dr the staff was present in many medieval manuals as both a means of self-defense, and as a training platform for other unambiguous battlefield weapons.

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You can feel free to search around in the Wiktenauer website for specific manuals, including Paulus Hector Mair, Joachim Meyer, Fiore dei'Liberi, and George Silver.

Much of the context for the culture of violence in Germany comes from, as abovestated, B. Ann Tlusty's wonderful The Martial Ethic in Early Modern Germany.

The translation of Meyer's words above come from Jeffrey Forgeng's translation of Meyer's 1570 Kunst des Fechtens or "The Art of Combat."

More info on the dusack can be found here: The Dussack - A Weapon of War