The development of a convective flow after sudden heating of a vertical wall that laterally bounds a layer of ethyl alcohol in a system consisting of vertical layers of alcohol and air, separated by a thin metal partition, is studied numerically in a conjugate problem statement. The equations of thermogravitational convection in the Boussinesq approximation, written in variables of temperature, vortex, and stream function, are solved by the finite element method. The development of unsteady hydrodynamic and thermal boundary layers on all four vertical walls is studied. The temperature fields in liquid, gas and in a vertical partition are calculated. The features of the development of the spatial form of the flow and unsteady conjugate convective heat transfer between the layers of liquid and gas affect significantly the unsteady temperature fields and temperature gradients in a thin metal partition. The maximum temperature gradients in the partition appear at the initial stage of flow development.