The most fascinating feature of certain two-dimensional (2D) gapless quantum spin liquid (QSL) is that their spinon excitations behave like the fermionic carriers of a paramagnetic metal. The spinon Fermi surface is then expected to produce a linear increase of the thermal conductivity with temperature that should manifest via a residual value ($kappa_0/T$) in the zero-temperature limit. However, this linear in T behavior has been reported for very few QSL candidates. Here, we studied the ultralow-temperature thermal conductivity of an effective spin-1/2 triangular QSL candidate Na$_2$BaCo(PO$_4$)$_2$, which has an antiferromagnetic order at very low temperature ($T_N sim$ 148 mK), and observed a finite $kappa_0/T$ extrapolated from the data above $T_N$. Moreover, while approaching zero temperature, it exhibits series of quantum spin state transitions with applied field along the $c$ axis. These observations indicate that Na$_2$BaCo(PO$_4$)$_2$ possibly behaves as a gapless QSL with itinerant spin excitations above $T_N$ and its strong quantum spin fluctuations persist below $T_N$.
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