I recall the BB episode in which the 6 kids tried out for a TV talent show, and I really recall the scene in which they were trying to secretly rehearse in the garage and Alice came along and saw them partially from under the partially closed garage "door" and asked them all what they were doing and the kids all started pretending that they were actually quickly sweeping the floor .
However looking at some of the links here (that leads to YouTube snippets of Brady Bunch scenes) it seems there were other episodes in which the 6 Brady Bunch kids performed a singing and dancing routine for TV talent show .

But I recall the 6 original , youngest cast members of the Brady Bunch once made an appearance on the Mike Douglas Show and they performed a singing and dancing routine and it did not seem to have anything to do with the episodes they did on the Brady Bunch .
It seemed that the cast were genuinely showcasing their other talents . I even recall reading in one of the teen magazines of the time (actually a few years before this Mike Douglas appearance) that the 6 original , youngest cast members of The Brady Bunch were said to have done a song and dance performance in some amusement park. From what I recall it also had nothing to do with the particular episode(s) in which the characters they played performed.

And all this was before The Brady Bunch TV variety show in which all but one of the cast members were gathered to portray the characters they originally played on the original Brady Bunch , but the strange twist was that they were all hosting a variety show like Donny and Marie or Sonny and Cher did. I guess most Brady Bunch fans (or evedn non-fans ) who were around in the 70s remember that.