Heat transfer through a liquid-vapor interface is a complex phenomenon and crucially relevant in heat-removal and cryogenic applications. The physical coupling among confining walls, liquid and vapor phases is very important for controlling or improving cooling rates or condensation efficiency. Surface modification is a promising route, which has been explored to taylor the heat transfer through confined two-phase systems. We use coarse-grained molecular-dynamics simulations to study the heat transfer through a nano-confined liquid-vapor interface as a function of fluid filling. We set up a stationary heat flow through a liquid-vapor interface, stabilized with the liquid in contact with a colder wall and a vapor in contact with a hotter wall. For these physical conditions, we perform extensive simulations by progressively increasing the number of fluid particles, i.e. the channel filling, and measure the fluid distribution in the channel, density, pressure and temperature profiles We also compare the heat flux behavior between a bare-surfaces nano-channel and others where the hot surface was coated with end-grafted polymers, with different wetting affinities and bending properties. We take extreme cases of polymer properties to obtain a general picture of the polymer effect on the heat transfer, as compared with the bare surfaces. We find that walls covered by end-grafted solvophylic polymers change the heat flux by a factor of 6, as compared with bare walls, if the liquid phase is in contact with the polymers. Once the liquid wets the coated wall, the improve on heat flux is smaller and dominated by the grafting density. We also find that for a wall coated with stiff polymers, the jump in heat flux takes place at a significantly lower channel filling, when the polymers' free ends interact with the liquid surface. Interestingly, the morphology of the polymers induces a ``liquid bridge'' between the liquid phase and the hot wall, through which heat is transported with high (liquid-like) thermal conductivity.