If you believe this site it states that nearly all of the sounds were original to Star Trek (at least one exception, down below).
From the sleevenotes: "The guiding genius behind this ... was Gene Roddenberry. Roddenberry recruited a hand-picked technical crew to create the incredible series ... Virtually all the sound effects were created exclusively for the television series ... principally by Jack Finlay, Douglas Grindstaff and Joseph Sorokin."
As to how it was made, Alan Howarth, who worked on every Star Trek film, has this to say about the Bridge sounds:
The bridge background of the 60's was electronic music with sonar beeps.
Ben Burtt, who worked on the new Star Trek 2009 movie, had this to say about the original sounds in an interview he gave.
Two things in the original Star Trek effects were revolutionary: Roddenberry had his team create lots of detail. Every room in the ship sounded different.
The other thing that was used a lot in the original show a lot was shortwave radio recordings and sounds off of transmissions and Morse code, things you can pick up in-between the dials on a shortwave radio.
It reads to the audience that you’re way the heck out at the edge of the universe, barely in contact. Things are far away: there’s these disembodied sounds that are being transmitted back and forth
From a different interview with Supervising Sound Editor Mark Stoeckinger on Star Trek 2009.
If you listen to TOS’ (The Original Series) sounds you can get a good idea what was used to create some of those sounds and so we would make button pushes and electronics out of bird calls, phone rings, animals screams or comedy effects as the originals were.
At least one sound on the show was copied (or duplicated)
In the original series, the steady blast of the phaser was derived from the hovering sound of the Martian war machines made for the 1953 version of Paramount’s War of the Worlds. The original was made with tape feedback of an electric guitar and a harp.