A charge $$q$$ is placed at the center of one of the surface of a cube. The flux linked with the cube is:

Charge q is at the center of one face of the cube.

Imagine attaching an identical cube to this face so the charge lies on the common face center, which becomes inside a rectangular box made of two cubes.

Now the charge is at the center of this combined box.

By Gauss law, total flux through the combined closed surface is

$$\Phi_{\text{total}}=\frac{q}{\varepsilon_0}$$

The two cubes are identical and symmetric, so flux splits equally between them.

Hence flux linked with one cube is

$$\Phi_{\text{total}}=\frac{q}{2\varepsilon_0}$$