A monitor will usually have an input selector switch.
Make certain that the switch on the monitor is set appropriately(dvi/vga/hdmi) and matches the graphics card connector.
Your graphics card should be able to handle any two of the possible output connections at the same time.
It looks like your motherboard also supports video output.
To use the motherboard outputs, make certain that the motherboard video chipset drivers have been installed.
If you are using windows 7, it will have no problem using multiple graphics drivers.
Some older motherboards disabled the integrated adapter when it detected a discrete graphics card.
And... I would favor the DVI connections over the vga connectors.
What kind of monitors are you trying to connect? Do either of them Support DVI connection to the monitor without the DVI to VGA adaptor? What is the model and manufacturer of the Video card?
In order to run two monitors on that card you will need to connect both video cables to the Radeon HD 6850. Using the motherboard video is not necessary and may not even be supported by the motherboard with a discrete graphics card installed. It will work best if you can connect each monitor to a digital video signal such as DVI, HDMI, or Display Port without using any adaptors. VGA is an Analog Video standard and it is most likely not supported on the second DVI connector.
One is BenQ monitor and other is HP w1907v. BenQ monitor supports DVI connection to the monitor, but HP w1907v doesn't support it. I think video card model and manufacturer is AMD Radeon HD 6800, if im looking from the right spot.
Well in that case, you need to hook up the second monitor using a DVI-DVI cable and plug that into the second port on the video card. Go to your local Computer store and buy a stadard DVI video cable. Most Radeon HD 68X0 series cards should support dual monitors in that configuration.