In this paper, we extend the discrete unified gas kinetic scheme (DUGKS) to simulate conjugate heat transfer flows. The conjugate interface that separates two media with different thermophysical properties is handled by a ghost-cell (GC) immersed boundary method. Specifically, fictitious cells are set in both domains to implement the continuity conditions of both temperature and heat flux at the conjugate interface. Information at the ghost cells is unavailable and should be reconstructed from the interface and neighboring field. With interface constraints involved in the reconstruction, the conjugate condition can be incorporated smoothly into the process of solving the full flow field. Those features make the present GC-DUGKS capture the conjugate interface, without introducing an additional term or modifying the equilibrium distribution function as in previous studies. Furthermore, the Strang-splitting scheme is applied for convenient handling of the thermal force term in the governing equation. Simulations of several well-established natural convection flows containing complex conjugate interfaces are performed to validate the GC-DUGKS. The results demonstrate the accuracy and feasibility of the present method for conjugate heat transfer problems.