We have studied the lattice dynamics and lattice thermal conductivity of NaCoO2 intercalation material with first-principles hybrid density functional methods. The lattice thermal conductivity has been obtained using linearized Boltzmann transport theory and the contributions to the lattice thermal conductivity have been analyzed in detail. The results obtained for NaCoO2 have been systematically compared with LiCoO2 to shed light on the effect of the alkali metal atom. The room-temperature in-plane lattice thermal conductivities within relaxation time approximation are 78 Wm-1K-1 and 46 Wm-1K-1 for NaCoO2 and LiCoO2, respectively. The respective room-temperature cross-plane lattice-thermal conductivities are 25.0Wm-1K-1 and 6.6 Wm-1K-1. The predicted lattice thermal conductivities for fully alkali-occupied single crystals are clearly larger in comparison to the experimental values obtained for single-crystal NaCoO2 and polycrystalline LiCoO2. Analysis of the lattice thermal conductivity reveals that the differences between NaCoO2 and LiCoO2 can be explained by significantly shorter phonon lifetimes in LiCoO2.