There are ways to record the sound coming through your sound card. It could easily be done, but if it was anything other than you recording for your own personal use (like, if you put the sound on the internet to share) you would very quickly run up against CG's IP protections and you wouldn't want to be on the wrong end of that lawsuit.
Why not just look up how to record your own device's sound output, then use a free sound editor to cut it down to what you want? If you're not undercutting their IP, I doubt (though I could be wrong) that they would mind you having a conversation-worthy unique text alert sound (which, of course, they would hope others would ask you about).