A line with a slope angle of $45^{\circ}$ has a slope of $1$ (since $\tan 45^{\circ} = 1$). The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\frac{y_2 - y_1}{x_2 - x_1}$. Therefore, we can set up the following equation:

$$\frac{m - 4}{3 - 2} = 1$$

Solving this equation for $m$ gives:

$$m - 4 = 1 \times (3 - 2)$$
$$m - 4 = 1$$
$$m = 5$$

So, the value of $m$ is $\boxed{5}$.