You know, as in "zzz"
The Oxford English Dictionary has these two entries:
4. a. Used (usually repeated) to represent a buzzing sound; also conventionally representing the sound of snoring. Hence Z-ing vbl. n., and as v. intr. to make such a noise or noises.
1852 H. D. Thoreau Jrnl. 15 June (1997) V. 96 The dry z-ing of the locust is heard.
1884 R. W. Buchanan New Abelard i, The bats were seen flitting with thin z-like cry high up over the waterside.
1893 R. Kipling Many Inventions 103 The oars rip out and go z-zzp all along the line.
1902 S. E. White Blazed Trail ii. 11 The rhythmical z-z-z! z-z-z!_[of the saw].
1909 H. G. Wells Tono-Bungay i. ii. 67 He had a way of drawing air in at times through his teeth that gave a whispering zest to his speech. It's a sound I can only represent as a soft Zzzz.
1909 H. G. Wells Tono-Bungay iii. ii. 326 He meditated for a time and Zzzzed softly.
1924 Dial. Notes 5 259 Z-z-z (buzzing, or snoring).
1951 Blue Book Mag. Jan. 25/1 A spark danced between two terminals, a filament snake spat an irate, ‘Zzzt!’
1966 L. Cohen Beautiful Losers i. 16 Hiccup, jerk, zzzzzz, snort.
1967 V. C. Welburn Johnny so Long ii. i. 46 Lola: (makes buzzing noise) Zzzzzzz.
1975 New Yorker 21 Apr. 36/3 David sits in the chair, puts his arms on the armrests, presses his neck against the back of the chair, and moves his feet together. ‘Zzzz,’ he says, and his head falls forward.
1976 Cambridge Independent Press 16 Dec. ii. 3/2 The zzzzz-noise of the electric hare gliding past the opening traps grabs everyone's attention.
1983 Private Eye 4 Nov. 6/2 Once you have hit on a commercial product you just go on producing more of the same, over and..zzzz..over and..zzzz..over and..zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz.
1984 Wall St. Jrnl. 9 Oct. 28/2 We suspect public interest..more nearly resembles a cartoonist's depiction of a man sawing wood—ZZZZZZZZZ.
1984 Oxford Star 29–30 Nov. 19/3 Zzzzing off for forty winks on a regular basis may not sound much like Action Man stuff, but for Alex Gardner it's the most exciting part of the day.
b. In colloq. phr. to catch some z's and varr., to get some sleep (where z represents the sound of snoring). U.S. Pronounced /ziːz/ in the U.S.
1963 Amer. Speech 38 174 An onomatopoetic construction reported four times is get some Z's... Variants occurring once were: bagging Z's, copping some Z's, cutting Z's, and knocking out Z's.
1973 A. Dundes Mother Wit 238 Got to go..cop me some z's.
1977 C. McFadden Serial (1978) xxxii. 71/1 All Harvey wanted..was to pop his Sominex and catch a few z's.
So, it seems like people have been using the letter Z to represent snoring since the 1920s at least, and the expression "get some Zs" at least since the 60s.
The first cite for "get some Zs" was an article in the journal American Speech by Alan Dundes and Manuel R. Schonhorn " Kansas University Slang: A New Generation" (JSTOR link), if you're curious.
Generally the OED staff are good at finding the earliest attestation of something; though obviously people tend to use a term for a period of time before it gets written down for posterity.
EDIT: I came across a claim that has found even earlier uses of Z for sleeping; The Straight Dope writes:
Once again my assistant Una stepped into the breach. Searching for the letter Z in the world’s databases turned up a considerable number of false positives, but by and by she found an instance of Z = snoring in the humor section of the January 1919 Boy’s Life, the Boy Scout magazine. Pushing on, she found the Krazy Kat comic strip of May 28, 1916, in which a sleeping bear emitting Z’s is awakened when Ignatz the mouse playfully chucks a rock at its head.
It soon became clear comics were the principal Z vector. In the Katzenjammer Kids strip of February 16, 1913, the sleeping Captain is generating b-z-z-z’s and z-z-z’s prior to having his rocking chair pulled over backwards by the disrespectful Kids opening the door.
In the November 17, 1907, edition of the comic strip The Fineheimer Twins, a blatant Katzenjammer knockoff, Una found a peg-legged man producing a whole alphabet of sounds while sleeping, including “g-r-r-k-k-k-k,” “z-z-z-c-r-r-k-k-k-k,” and plain old “z-z-z,” until a fishbowl is upended on his head.
But the ur-instance of Z, or at least the earliest that's come to light, was turned up by Sam Clemens of the Straight Dope Science Advisory Board. It was again from the Katzenjammer Kids, and again featured the snoring Captain, this time suspended in a hammock, unaware he’s inventing an enduring comic strip trope. The unimpressed Kids trim his beard with a push mower, then end further Z-ifying by cutting the hammock’s ropes. Date of these epochal events: August 2, 1903.
And credit to tumblr user robertloerzel for tracking down that initial comic from 1903. (imgur mirror).
EDIT 2: I was curious about the Boy's Life citation, since it was the earliest one they mentioned not in a comic usage, so I tracked it down. There it's used as the title/header of the joke. (Source: Google Books)
Z-z-z-z-z-z-z-z-z
Tenderfoot: "Do you know the scientific name for snoring?"
Eagle Scout: "Uh, no."
Tenderfoot: "Sheet music."