The sleeve of a drill press or lathe headstock in which the spindle is mounted. On the vertically mounted drill-press, the quill moves up and down; on the horizonitally-mounted drill press, and the lathe, the quill moves back and forth. The quill -- which does not rotate -- serves a cylindrical housing to keep the spindle secure as it rotates. The quill usually has teeth in which a pinion engages to control the movement.
Source: adapted from Home Craftsman 4 January-February 1935 page 124.
The hollow steel mandrel of a seal-engraver's lathe, into which the engraving tools are fitted.
Knight American Dictionary of Mechanics 2081/2
The quill is of steel, about 2 inches long and 
inch in diameter.
A hollow sleeve rotating in bearings which is used to transmit the drive from a motor to a concentrically-mounted axle.
1910 Engineering 12 Aug. 246/3
A gearless concentric motor for each driving-axle is mounted on a quill flexibly connected to the driving-wheels.
1930 Ibid. 6 June 722/1 Two new types of drive had been developed... The first consisted of a geared quill surrounding the driving axle and carrying two crankpins, the latter being connected by a flexible linkage to two crankpins on the driving wheels.
1968 D. W. & M. HINDE Electr. Traction Systems & Equipment ii. 32/2
A certain amount of experimental work has been carried out with the motor armature shaft of the hollow or quill pattern.
1975 Bram Downs Manuf. Technol. vii. 208
The spindle rotates in the quill to provide the rotary motion for cutting tools.
Source: Oxford English Dictionary
The head is fastened to the upper end of the column. It has a central shaft or spindle, which rotates in a sleeve called the quill....
... The quill, together with the spindle, is moved up or down either by a single bar or by a wheel having three or four spokes or bars with a ball on the end of each. The quill has a stroke or travel from 3" to 4", Fig. 3. It is automatically re-turned to its normal position by a spring. Machines for production work are equipped with a power feed or treadle arrangement.
Source: Milton Gunerman, How to Operate Your Power Tools New York: Home Craftsman Publishing Corp., 1950, page 41; the image on the left comes from the same page.
The original definition was couched in rather confusing terms, so I tried to clean it up a bit. I admit greater interest in the origin and background history of "quill" than most terms of woodworking. cabriole and Windsor chair are other woodworking terms that are different, perhaps because of the history underlying their coinage.
Quill, in particular, stands out. "Hollow sleeve that houses a rotating spindle" seems simple enough, I guess, until you begin thinking about how the term quill was applied to the concept of a sleeve for housing a spindle.
After a fruitless search in etymology dictionaries, and the like, I have concluded that no one has speculated on the origin of the meaning of quill as a sleeve. I have a theory though, even if it sounds far fetched. Think of a old timey pen, fashioned out of a feather. The feather's central component, (ie, the harder, central element that grows out of a bird's body and gives support to the feather) is hollow, and with its hollowness, and that it can be sharpened, means that feathers can be used as crude pens. This fact was discovered many generations ago.
(Not a physicist, but I know a little about how liquids such as ink behave in certain situations -- here I am thinking of capillary action -- the quill of the feather acts as a "sleeve" for the ink. Could the first attempt at fashioning a quill for the first drill press had a feather's quill in mind while inventing the mechanism? Far fetched? Probably! But stranger things have happened in giving meaning to terms. I'll let this stand until some better explantion comes along.)
[Patent] 34,335.—J. F. Sargent (assignor to Elmer Townsend), of Boston, Mass., for Improvement in Machines for Pegging Boots and Shoes:
I claim as a new machine the combination of the mechanism for operating the awl, peg driver, and for feeding the work, with the mechanism for
cutting and feeding the peg work, all being arranged compactly in the frame, A, or its equivalent, and operated by the cams and levers, arranged substantially as and for the purposes described.
I also claim the pendulum or stying piece, H, having the awl and peg-driver carrier. L, the throat piece, b, the peg box, W, the pointing mechanism and pegwood feeder, arranged and applied thereto, or connected therewith, as set forth, in combination with so applying such pendulum to a quill or sleeve, F, disposed on the driving shaft, B, or on a stud or arm arranged just above or below the same, that s whole may be caused to operate together in manner and for the purpose set forth.
I also claim combining and arranging with a vibrating peg box and peg-wood feeder, constructed as described, a stationary knife, whereby the pegs are severed from the peg strip, in manner as set forth.
I claim so constructing and applying the throat piece or block to the pendulum, H, as to have no vertical movement, in combination with so forming and applying the retainer that it may have a short vertical movement, whereby the two are made to operate together in manner as set forth.
Source: Scientific American, New Series, Volume 6, issue 8, February 22, 1862, page 125.
The text in the box below comes from R J DeCristoforo's first Shopsmith "Bible", the 1953 edition of Power Tool Woodworking for Everyone, a woodworker's manual treasured by owners of the original Shopsmith model 10E and 10ER. (Click here for more about DeCristoforo.)
Two years later, when the Shopsmith Mark IV his the market, DeCristoforo updated his first edition to reflect the changes the Mark IV contained. And in 1955 edition of Power Tool Woodworking for Everyone, expands on his explanation of the mechanics of the quill on the drill press and othe similar woodworking tools with headstocks.
(By his own admission, DeCristoforo wrote both the 1953 and 1955 Shopsmith manuals for woodworking tools in general, not just for the Shopsmith. Notice, though, that on the Shopsmith, the quill sleeve is activated by a lever, not a crank or a wheel, as indicated in the Lammey fragment, below DeCristoforo. )
.... The quill (the sleeve in which the spindle rotates) has a feed of 4 inches and is returned to its starting position by a coiled spring enclosed in a case on the switch side of the head-stock. [See details of DeCristoforo's explanation of the quill in the image of the anatomy of the Shopsmith headstock on this page.]The factory-adjusted spring can be re-adjusted to strengthen or weaken the pull with which it returns the quill. Normally, it should return smoothly and without shock. The quill assembly is moved downward by a feed lever which actuates a rack and pinion gear. One or more feed levers can be set in feed knobs on either side of the headstock, and a little more than one complete revolution of the knob extends the quill the maximum 4 inches. The quill lock lever will secure the quill in any extended position. The stroke may be set for a predetermined depth by using the stop nuts on the depth stop rod, which is secured to the gauge collar. Direct readings may be taken from the indicator fastened against the spring housing on the pinion shaft....
Source: R J DeCristoforo Power Tool Woodworking for Everyone New York: McGraw-Hill, 1953, page 78 [link is to 1984 edition]
Quill are activated by both levers and cranks/wheels:
Slide the tailstock back on the bed to the required distance to take the stock between the centers. Place the left end of the stock on the spur center and raise the opposite end so that the point of the dead center will enter the center hole in the stock when the tailstock quill is run forward by turning the handwheel or crank.
Source: W. Clyde Lammey, Power Tools and How to Use Them Chicago: Popular Mechanics Press, 1950, page 59.
Sources: Home Craftsman 4 January-February 1935; R J DeCristoforo Power Tool Woodworking for Everyone New York: McGraw-Hill, 1953.