I am fairly sure Queen performed Bohemian Rhapsody on TV sometime between (I would guess) 1982 and 1987, probably leaning towards earlier? This most likely would have been a national broadcast. My mom didn’t like the grim stuff the opening lyrics described.
What show or other circumstance would they have been performing in?
They performed the song in nearly every concert, but I don’t think they ever performed the operatic portion on stage. Some times they used the recording of that section, sometimes they just performed the first part and guitar solo (like in Live Aid).
Could you be thinking of the video which if I remember correctly has concert footage in it (clips of it were in the Wayne’s World version of the video).
This is certainly what the OP made me think of. Live Aid was in the summer of 1985 and was internationally broadcast on MTV. Queen were one of many acts who performed, and their performance has become arguably the most iconic. Their set opened with “Bohemian Rhapsody”—not the full song, but enough that your mom would have heard the opening lyrics.
If memory serves (not first hand, but from recordings, and the film), Freddie didn’t perform the opening lyrics and came in at ‘mama, just killed a man’ whilst playing the piano at Live Aid.