Conjugate heat transfer (CHT) analyses are vital for the design of many energy systems. However, high-fidelity CHT numerical simulations are computationally intensive, which limits their applications such as design optimization, where hundreds to thousands of evaluations are required. In this work, we develop a modular deep encoder-decoder hierarchical (DeepEDH) convolutional neural network, a novel deep-learning-based surrogate modeling methodology for computationally intensive CHT analyses. Leveraging convective temperature dependencies, we propose a two-stage temperature prediction architecture that couples velocity and temperature fields. The proposed DeepEDH methodology is demonstrated by modeling the pressure, velocity, and temperature fields for a liquid-cooled cold-plate-based battery thermal management system with variable channel geometry. A computational mesh and CHT formulation of the cold plate is created and solved using the finite element method (FEM), generating a dataset of 1,500 simulations. Our performance analysis covers the impact of the novel architecture, separate DeepEDH models for each field, output geometry masks, multi-stage temperature field predictions, and optimizations of the hyperparameters and architecture. Furthermore, we quantify the influence of the CHT analysis' thermal boundary conditions on surrogate model performance, highlighting improved temperature model performance with higher heat fluxes. Compared to other deep learning neural network surrogate models, such as U-Net and DenseED, the proposed DeepEDH architecture for CHT analyses exhibits up to a 65% enhancement in the coefficient of determination $R^{2}$. (*Due to the notification of arXiv "The Abstract field cannot be longer than 1,920 characters", the appeared Abstract is shortened. For the full Abstract, please download the Article.)