1. Set the side length of the octahedron to 1 for simplicity.
- The volume of a regular octahedron with side length $a$ is given by:
$$V_O = \frac{\sqrt{2}}{3} a^3$$
- For $a = 1$:
$$V_O = \frac{\sqrt{2}}{3}$$

2. Determine the height of the octahedron.
- An octahedron can be divided into two pyramids with a square base.
- The height of each pyramid can be found using a 45-45-90 triangle:
$$\text{Height} = \frac{\sqrt{2}}{2}$$

3. Find the side length of the cube.
- The vertices of the cube are the centers of the faces of the octahedron.
- The center of each face of the octahedron is the circumcenter of an equilateral triangle.
- The height of an equilateral triangle with side length 1 is:
$$\text{Height} = \frac{\sqrt{3}}{2}$$
- The circumcenter is $\frac{1}{3}$ of the way up the height of the triangle:
$$\text{Distance from base to circumcenter} = \frac{\sqrt{3}}{6}$$
- The height of the pyramid above the circumcenter is:
$$\text{Height above circumcenter} = \frac{\sqrt{3}}{3}$$

4. Calculate the side length of the cube.
- The side length of the cube is the distance between two adjacent face centers of the octahedron.
- This distance is the hypotenuse of a right triangle with legs $\frac{\sqrt{2}}{3}$ and $\frac{\sqrt{3}}{3}$:
$$\text{Side length of cube} = \sqrt{\left(\frac{\sqrt{2}}{3}\right)^2 + \left(\frac{\sqrt{3}}{3}\right)^2} = \sqrt{\frac{2}{9} + \frac{3}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}$$

5. Calculate the volume of the cube.
- The volume of a cube with side length $s$ is $s^3$:
$$V_C = \left(\frac{\sqrt{5}}{3}\right)^3 = \frac{5\sqrt{5}}{27}$$

6. Find the ratio of the volumes.
- The ratio of the volume of the octahedron to the volume of the cube is:
$$\text{Ratio} = \frac{V_O}{V_C} = \frac{\frac{\sqrt{2}}{3}}{\frac{5\sqrt{5}}{27}} = \frac{\sqrt{2}}{3} \cdot \frac{27}{5\sqrt{5}} = \frac{27\sqrt{2}}{15\sqrt{5}} = \frac{27\sqrt{2}}{15\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{27\sqrt{10}}{75} = \frac{9\sqrt{10}}{25}$$

7. **Simplify the ratio and find $m+n$.**
- The simplified ratio is:
$$\frac{9}{2}$$
- Therefore, $m = 9$ and $n = 2$:
$$m+n = 9+2 = 11$$

The final answer is $\boxed{11}$.