The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved.
However all the papers that I have seen dealing with this problem or variants of this problem say things like "It is a longstanding conjecture that a transitive projective plane is Desarguesian" or "It has been conjectured that ..." and none of them actually say who made the original conjecture.
I've looked in Kantor's papers on flag-transitive planes, Dembowski's book on Finite Geometries, Ostrom and Wagner's paper proving that planes with doubly transitive groups are Desarguesian and Higman and McLaughlin's paper on ABA groups.
So is the conjecture folklore? Or can anybody point me to an explicit reference?
EDIT: Question has been up for a few days without answer so I'm giving up and assigning the result to "folklore". In the meantime, I've written a blog post about it for posterity: http://symomega.wordpress.com/2011/06/03/an-elusive-conjecture/