"After the 1974 London Conference, with its strong confirmation of the quark model, a general change of view developed with regard to the structure of hadrons. [...] the quark structure of hadrons became the dominant view for developing theory and planning experiments. A crucial element of this change was the general acceptance of QCD, which eliminated the last paradox, namely, why are there no free quarks ? The conjectured infrared slavery mechanism of QCD provided a reason to accept quarks as physical constituents without demanding the existence of free quarks. The asymptotic freedom property of QCD also provided a ready explanation of scaling [...]. There were a number of other important experimental results reported in1974 and in the latter half of the decade that provided further strong confirmations of the quark model. Among these were the discovery of charmonium, and its excited states [...]. The quark model, with quark interactions described by QCD, became the accepted basis for the structure of hadrons."
Jerome Friedman, "Deep-Inelastic Scattering and the Discovery of Quarks", in "The Rise of the Standard Model", ed. L.Hoddeson, L.Brown, M.Riordan, M.Dresden, Cambridge UP 1997, p.584.
So, I use to teach my students that it was the discovery of the J/ψ at SLAC and Brookhaven what tilted the balance, but Friedman says people were already convinced of the existence of quarks before that... I guess the truth is in the middle.