Pseudo rounded cube in OpenGL
If silhouette is not important, you could bend the shading normal near edges. 

Since it's a cube you can probably compute the normal procedurally in the shader based off of the uv. I.e. if uv is in range [0,0.1] or [0.9,1.0] start bending the normal towards horizontal. 

If silhouette is important, drawing a rounded cube model with vertex normals would make the most sense. There are cube models which have the beveling to look like a rounded cube.