In the paper we develop, in the CGC approach, a new mechanism for quarkonia production in events with high multiplicity of the soft produced particles. In this mechanism both J/ Ψ and soft hadrons are produced at short distances r∝1/Qs, where Qs is the saturation scale, which is considered to be proportional to the multiplicity of soft hadrons. Using this approach, we analyze the vigorously growing multiplicity dependence of quarkonia production, which was recently discovered by the STAR and ALICE collaborations. We demonstrate that this result presents a strong evidence in favor of multigluon fusion mechanisms of the quarkonia states. We analyze in detail the contribution of the 3-gluon fusion mechanism and demonstrate that it describes correctly all the available experimental data on the multiplicity dependence of J/ ψ meson production. We also make predictions for other quarkonia states, such as ψ(2 S) and Υ(1 S) , and find that the multiplicity dependence of these states should be similar to that of the J/ ψ meson. Finally, we discuss an experimental setup in which the very strong multiplicity dependence could be observed. This observation would be a strong evidence in favor of the CGC approach.