There are some diagnostics software that you may be able to download from the internet, that could detect your graphics card. However, a lot of these type of diagnostics, actually rely on the drivers already being installed, but, some do actually scan the graphics chip to determine its identity.
The other way, is, if you feel confident in taking your computer apart, then you can look at the grahpics card, and on some of the chips, varies with each card, should be a name. From this name, it is possible to get the appropriate drivers for the chip.
Many chip manufactures make chips for different card manufactures, who in turn, add their own unique features to the card, but, the basic drivers for the chips, would suffice just to get the correct resolution and amount of colors. Of course, it would be much better if you could get the actual card drivers, which will have the chip drivers, and any extra features that may have been included in the card.
Note: ONLY TAKE YOUR COMPUTER APART IF YOU ARE SURE! YOU KNOW WHAT YOU ARE DOING!
Hope this helps.