In Oedipus Rex, suppose King Oedipus had taken Jocasta’s advice and given up on his quest. How might the play have had a different ending?

This is an interesting question because I guess it depends
on a lot of other factors that could somehow influence the outcome. Certainly we can say
that all those who give Oedipus evidence in his quest for the truth do not do so
willingly--Oedipus needs to threaten, cajole and force them to reveal what they know.
Therefore if Oedipus had abandoned his quest it is possible that he could have got away
with it. However, imagine the scene - Thebes is suffering gravely as ever more people
die from the mysterious illness. If Oedipus hadn't found "the truth," then more and more
people would have died and perhaps an Oracle could be found who wouldn't be so shy of
speaking the truth. My own personal feeling is that this is a case where the "truth will
out" whatever happens, and Jocasta, even were she to be successful, would only put of
the inevitable revelation that, at least on some level, she herself is aware
of.