Maybe you've already seen this: a description of the episode The Ballad of Molly Turgis, but it doesn't mention the song.

I seem to recall, from my days of watching TV sitcoms (going back to "Our Miss Brooks" and "Amos 'n' Andy"), that "Gertrude" was one of those old-fashioned names that was thought to automatically conjure up an image of a stodgy old lady, and was frequently used for that purpose. I'd guess the song was made up for the show—unless they took an existing song and modified it.