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Of American-English origin, the noun air guitar designates an imaginary guitar played by miming, especially to rock music.

—Cf. also the plural noun air quotes, which designates a notional set of quotation marks gestured by a speaker’s fingers in the air.

The earliest occurrences of the noun air guitar that I have found are from Rock’n’roll imagination unleashed, by David Klein, Tallahassee Democrat staff writer, published in the Tallahassee Democrat (Tallahassee, Florida, USA) of Sunday 12^th November 1978 [page 3E, columns 1 & 2]:
—Note: This article also contains the noun air band, designating a pretend rock band whose members mime the action of playing imaginary instruments:

The part of rock’n’roll you might call pure showmanship [is] the part that Florida State University students gathered to celebrate in their union courtyard Monday.
It might have been a celebration of the first event of its kind ever held in America.
It was lableled [sic]: “Air Guitar Contest.”
You ask, what is an “air guitar?”
Well … an air guitar is a guitar you pretend to play when you don’t really have one—and don’t know how to play one anyway.
It’s like when you’re alone in your dorm room, you know, and you’ve got the stereo turned up real loud, and you’re just kinda rockin’ out all by yourself, and your fingers start gently rippling those shadowy electrified strings, keeping up with ol’ Neil Young as he pulls hot licks screaming from his guitar, matching noise with Pete Townshend as he tears his way through a progression of power chords, even hanging in tight with Jimi himself, ripping out notes line for line against the smokiest velvet chords Hendrix can throw at you.
But it’s all a game of pretend, you understand.
[…]
[…] Let’s return to the FSU courtyard […].
Joint sponsors, CBS records […] and the campus Leisure Program Office were ecstatic at the turnout—hundreds of students.
[…]
Each contestant that afternoon would sing two songs, one from a new album that CBS is promoting, and one of his own choosing.
The winner of this epochal “air guitar” contest would be chosen by audience applause.
The prize: 25 CBS records.
What an event!
Here at long last the ever-present—though never-mentioned—air guitar is finally exiting from the closet.
Johnny Elijah, the first contestant, came screaming across the stage, a cord hooked to his belt on one end and to an amplifier on the other. The crowd cheered him on as he threw off his coat, kicked one leg high and got down to business.
Behind him was a sound system loaded up with 120 watts per channel, pouring out the music for his air guitar. […]
[…]
The next act was a whole “air band,” complete with drummer, guitarist, bass player, keyboards player and vocalist.
[…]
This band brought screams from the crowd when the guitarist and bass player leaned on each other, back to back, wiggling their fingers furiously at their waist and grimacing with the concentration of this joint pretend musical effort.