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Heavy quarkonium production is an excellent tool to test both perturbative and non-perturbative QCD, as perturbative QCD can describe the heavy quark production process, while the formation of the quarkonium bound state involves non-perturbative aspects. In ultrarelativistic heavy-ion collisions, a deconfined state of QCD matter, made of free quarks and gluons, called quark-gluon plasma (QGP) is expected to be formed. To probe such an environment, the study of quarkonium production is an important tool as the heavy quarks are produced during the initial stage of the collision and experience the entire medium evolution. Surprisingly, in small colliding systems as pp and p--Pb, QGP-like behaviours are observed when selecting high multiplicity events. The physics interpretation of these behaviours remains unclear. However, multiparton interaction is one of the main promising scenarios to explain such observation. Studies of charmonium yields as a function of the event charged-particle pseudorapidity density in pp and p--Pb collision allow one to probe multiple parton interactions in an indirect way. These proceedings present the measurements of quarkonium yields normalised to their average values as a function of the charged-particle multiplicity in pp collisions at $\sqrt{s}$ = 13 TeV and in p--Pb collisions at $\sqrt{s_{\rm NN}}$ = 8.16 TeV, performed by the ALICE experiment at the LHC. The corresponding results for the $ψ\rm{(2S)}$-to-J/$ψ$ ratios as a function of charged-particle multiplicity are also shown. In addition, J/$ψ$ pair production measurement in pp collisions at $\sqrt{s}$ = 13 TeV is discussed.