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Investigating the exotic quantum phenomena and the related ground state of the frustrated quantum magnets is a focus of condensed matter physics  over the past years. While the realization of quantum spin liquid (QSL) without conventional magnetic ordering even down to 0 K is a challenging task, the Kitaev model is a prominent example with an exactly solvable spin model. The Kitaev materials so far have been focused on 4d/5d based systems with effective spin-1/2, such as H₃LiIr₂O₆, α-Li₂IrO₃, α-Na₂IrO_3, and α-RuCl₃, which exhibit zigzag antiferromagnetic (AFM) ordering at zero magnetic field owing to non-Kitaev interactions. Recently, the theoretical studies propose that the 3d-cobalt honeycomb magnets with effective spin-1/2 can be another playground to realize the Kitaev model. The non-Kitaev terms have been predicted to be almost vanishing with the small trigonal crystal fields acting on Co²⁺ ions, which makes the cobaltates as one good system for Kitaev physics. We studied the potential candidate Na₂Co₂TeO₆ and performed magnetization, heat capacity, high-field electric spin resonance, and inelastic neutron scattering (INS) measurements. A complicated magnetic field-temperature phase diagram was demonstrated with a zigzag AFM state and spin gap was observed in the spin dynamics. A quantum spin disordered state is induced by an ab-plane magnetic field between 7.5 and 10.5 T like the field-induced QSL in α-RuCl₃. Through detailed theoretical simulations on the INS spin-wave spectrum, we could obtain different ratios of the Kitaev interaction and the Heisenberg exchange. Hence, we anticipate our work will generate significant interest: it will trigger new experimental efforts for other realizations of magnetic field inducing the exotic quantum states and rejuvenate theoretical and computational efforts to understand the elusive dynamics in the Kitaev model for frustrated quantum magnetism.