In heavy-ion collisions at the LHC, the ALICE Collaboration is studying Quantum Chromodynamics (QCD) matter at very high energy density where the formation of a Quark Gluon Plasma (QGP) is expected. Quarkonium production is an important probe to characterize the QGP properties. High precision data in pp collisions provide the baseline of the Pb–Pb measurements and data in p–Pb collisions serve to quantify the contribution of initial and/or final state effects, related to cold nuclear matter. Since 2010, the LHC provided Pb–Pb collisions at sNN=2.76 TeV, pp collisions at various energies and in 2013 p–Pb collisions at sNN=5.02 TeV. ALICE measures quarkonium production from zero transverse momentum in the dimuon channel at forward rapidity and in the dielectron channel at mid-rapidity. This proceedings presents the new results on inclusive production of J/ψ, ψ(2S) and ϒ performed in p–Pb collisions and on the pT dependence of inclusive J/ψ in Pb–Pb collisions. The contribution of J/ψ from B hadrons to the inclusive production in Pb–Pb is also discussed. Finally, the p–Pb measurements allow an estimation of the contribution of the cold nuclear matter effect to the Pb–Pb measurements and this is also reported.