Solution:

Let $Q = \left(\frac{5}{8}, \frac{3}{8}\right)$ and $P = (x, y).$ The slope condition $\frac{y - 3/8}{x - 5/8} \ge \frac{1}{2}$ becomes $y \ge \frac{x}{2} + \frac{1}{16}$ when $x \gt \frac{5}{8},$ and $y \le \frac{x}{2} + \frac{1}{16}$ when $x \lt \frac{5}{8}$ (multiplying by the negative quantity $x - \frac{5}{8}$ reverses the inequality).

For $x \gt \frac{5}{8},$ the region above the line $y = \frac{x}{2} + \frac{1}{16}$ inside the square is a trapezoid with parallel vertical sides of lengths $\frac{5}{8}$ (at $x = \frac{5}{8}$) and $\frac{7}{16}$ (at $x = 1$) and width $\frac{3}{8},$ with area $\frac{3}{8} \cdot \frac{5/8 + 7/16}{2} = \frac{51}{256}.$ For $x \lt \frac{5}{8},$ the region below the line is a trapezoid with parallel sides $\frac{1}{16}$ (at $x = 0$) and $\frac{3}{8}$ (at $x = \frac{5}{8}$) and width $\frac{5}{8},$ with area $\frac{5}{8} \cdot \frac{1/16 + 3/8}{2} = \frac{35}{256}.$

The probability is $\frac{51}{256} + \frac{35}{256} = \frac{86}{256} = \frac{43}{128},$ so $m + n = 43 + 128 = 171.$