Hi and welcome to ExTS
Your m/board specs
http://www.gigabyte.lv/products/page/mb/ga-g31m-es2l_1x/
Your card specs:
http://www.nvidia.com/object/product-geforce-gtx-465-us.html
Your m/board has 1 PCIe x 16 slot which will take your new card.
Your new card is PCIe 2.0 compatible and from what I can tell from a quick search, your m/board supports PCIe 1.0 ( Perhaps others can confirm this ? )
If the m/board was produced pre 2007 then It will not support PCIe 2.0
If this is the case - your new card is backwards compatible.
In other words, it will work in your m/board but you may not get the full benefit of the card's features as PCIe 2.0 transfers data at twice the rate.
You don't say what PSU ( wattage ) you have.
If you do decide to go for this card then you may need to upgrade the PSU too.