In this paper, we develop the Fourier transform approach to study the Hyers-Ulam stability of linear quaternion-valued differential equation with real coefficients and linear quaternion-valued even order differential equation with quaternion coefficients. It shows that Fourier transform is valid to find the approximate solutions for quaternion-valued differential equations by considering their corresponding complex representation of quaternion-valued problems. Finally, two examples are given to illustrate the theoretically results.