The shows were What's My Line and I've Got a Secret. Palmer's What's My Line guest spot was in 1960, and in 1963 he was on I've Got a Secret.
The two game shows were very similar: Both brought guests on to be questioned by a panel of celebrities, whose job it was to guess either the occupation of the guest (in the case of What's My Line) or some interesting tidbit or amazing fact about the guest (in the case of I've Got a Secret).
It tells you something about the status of golf in 1960, and the media environment of the time, that Palmer - already a multiple major championshp winner - could appear before the celebrity panel, undisguised, on the premise that they would try to figure out what his occupation was.
Here are the two clips: