I see that this class can customize the style of the cube, such as its size, font color, etc.,
But it may not meet the requirements yet. If I want to achieve the effect shown in the picture below, are there any methods I can use in HOOPS communicator?
There is nothing preventing you from creating a nav-cube that looks like this in HOOPS Communicator, its just a bit of work. You would start by modifying the _createGeometry function in the nav-cube source code to include the additional faces or maybe model a cube like this in blender and load it into HC on startup. The additional sprites for the two extra buttons could either be SVG or billboarded geometry.