Is it possible to use a nVidia GPU to render graphics if the monitor is connected to the motherboard video port?

A program can access the dedicated graphics processor regardless of whether a display is attached to it, but additional software is needed to pass the output to the onboard GPU for display.

As the dedicated video card is still accessible to applications, the GPU could be used to render graphics and perform computations (like video transcoding and PhysX) as usual, but it won't be able to draw the output on the display unless you use a third-party program to send output to the onboard GPU. An example of such a program is Virtu MVP (now discontinued), which did this by intelligently abstracting onboard and dedicated GPUs so that applications see them as one device while taking full advantage of the unique features of each.

In the scenario you describe, by default the on board graphics would be used. The game could not switch to offload the rendering of its graphics to the NVidia card. However, if the software/game had a specific setting for such a thing then this could be possible.

For example, when doing Raytracing with Adobe After Effects, my primary cards are ATI Radeons. However, I have also an unused NVidia card which handles the raytracing. Graphic and Video Editing applications sometimes have certain settings where you can offload your computation needs to a graphics card that is not currently in use by your monitors.

Another example would be using the NVidia card as a Dedicated Physx Card. While this may increase some performance, it would be application specific if this could be taken advantage of (typically moreso in games).

Another example would be H264 Rendering with NVENC-export where you can pass a job to the NVidia card to handling the transcoding calculations.