is the intel quark capable of running an opencv webcam unity application for linux?
Most of the Quark parts are effectively small microcontrollers. For example, the Quark D2000 that you linked to runs at 32 MHz, and its working memory is limited to 8 KB of internal SRAM. This is completely inadequate for any kind of image processing, even if it had any way of driving a display output (which it does not).

The Quark X series (e.g, Quark X1000) runs at a more useful speed (400 MHz)  and supports external DDR3 memory. However, this series was released in 2014 and suffers from some serious CPU errata which make most standard Linux software unusable. If cost is a concern, I'd recommend that you look into alternative software that will run on an ARM SoC.