Hi AugReality,
Personally I have not tried to attach any graphic card to the PCIe slot, but sounds like an interesting project that I might try, however here I have some considerations that I think you will find useful.
According to the manual that you have provided http://www.neutronusa.com/ds.cfm/1336068 this PCIe card only supports (-O.S. support: Windows 7, Windows XP, Windows Server 2003, Windows Server 2008). Before you do any testing I would verify if the card is compatible with Linux an if it has drivers for it, as the Galileo currently supports Linux as O.S. Link to the Galileo FAQs https://communities.intel.com/message/207619
Another important fact that you need to keep in mind is power consumption, the Galileo board power adapter input must be 5V (7-12V are not supported), see the Galileo FAQs .
If you want to know about the values of consumption you can take a look at the Galileo Safety regulatory information document https://communities.intel.com/docs/DOC-21892, page 3, over there we have the values of the Galileo with full load, PCIe and other devices. I would like to point out the maximum amount that it would be 2A for a 10W total consumption. You should verify the PCIe device power consumption information of that card to avoid damage (burn) due to demanding power.
The PCI Express* mini-card slot, PCIe* 2.0 compliant features, works with half mini-PCIe* cards with optional converter plate and Provides USB 2.0 Host Port at mini-PCIe* connector
You can verify this information over the Galileo Datasheet https://communities.intel.com/docs/DOC-21835, page 6. I would consider this as well for the PCIe device that you will like to test and if you have some questions about the PCIe 2.0 specs, I found these informational pages that are very useful http://www.pcisig.com/news_room/faqs/pcie2.0_faq/ and http://en.wikipedia.org/wiki/PCI_Expres
Regards,
LDiego_Intel