f_Z(z) = \int_{-\infty}^{+\infty}f_Y(y)f_X(z+y)dy = \int_{-\infty}^{+\infty} \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(y-\mi)^2}{2\sigma^2}} * \frac{1}{4} dy = \frac{1}{4}