No it will not work with a VGA monitor.
If it is specifically marked as RS-232 (or with a string of one’s and zero’s), then it is a serial port for things like serial mice.
If it on the other hand, it is marked with Mon. or a rectangle, then it is a DE-9 connector for CGA or EGA monitors.
VGA monitors use a special D-sub 15-pin connector (DE-15F) and cannot be connected even with an adapter because the pinouts are completely different (because VGA is analog and works very differently from the older, digital CGA and EGA standards). You could try looking for a converter, but I for one have not seen or heard of any such thing, and would imagine it would be expensive anyway.
Your best bet is to check the local classifieds (especially online ones such as Kijiji or eBay Classified) to see if anyone nearby has a 9-pin monitor collecting dust in the closet. You could also put your own ad specifically asking for one.
Seeing your photo and the model of the motherboard, it is most definitely a serial port. The P4C800 does not have a 9-pin monitor connector (or even onboard VGA).
What you see is a generic bezel not specific to that motherboard. Cases used to come with several bezels of common variations, but because these days, motherboard backplanes come in a wide variety, they don’t and instead, motherboards come with a bezel specific to the board that can be snapped into a case. The one you see has both icons because it can be used for different motherboards and on some, instead of a serial port, there is a 15-pin VGA connector there (which has the same physical dimensions of a 9-pin serial port).
In summary, no, a serial port is not a video port and there is no way to use a monitor of any kind with it (even a 9-pin monitor, let alone 15-pin one). Since it does not have onboard video, you will need to install a video-card.