Neutron imaging is a powerful and nondestructive tool for testing materials in industrial and research applications. Compact accelerator neutron sources are gaining interest in neutron application techniques, such as Bragg edge transmission imaging. Delivering a high neutron flux with a narrow pulse width and suppressed photons at the sample position are fundamental factors for designing a neutron source for Bragg edge imaging. In this study, Monte Carlo calculations were performed to simulate a 40-MeV electron beam impinging on a cylindrical tungsten target. Different target moderator and reflector (TMR) geometries were investigated to produce cold neutrons, and their results were compared. Polyethylene (PE) and graphite were used as the moderator and the reflector, respectively. The structures and dimensions of the moderator and reflector were optimized using a Monte Carlo simulation with the PHITS-3.28 code. The effect of the PE moderator temperature on the cold neutron flux was investigated. The results showed that the optimum size of the PE at 77 K inside the reflector was 3  15  15 cm3 to achieve the wavelength resolution of 1.05% and the neutron flux of 1.16  104 n/cm2/s at 1000 cm from the target station by assuming the electron beam current of 275 µA. In addition, the FLUKA 4-2.1 code was used to calculate the neutron spectrum from the designed neutron production target at room temperature, and the results were consistent with the PHITS calculations. The neutron spectrum together with its pulse width from the designed TMR were used to simulate the Bragg edges of an -Fe sample, and it was concluded that the TMR is suitable for performing Bragg edge imaging.