There a few ways I can think of. Here are some ideas.
1. Turn on netflow switching, wait a while, and then type "show ip cache flow". Look for destination IP addresses that are not in your routing table.
2. Turn on IP accounting, wait a while, look in "show ip accounting" for destinations not in your routing table.
3. Turn off logging to the console and then turn on ip packet debugging (debug ip packet). This can be VERY hard on your router and could cause it to lock up. Use at your own discretion. However, if you don't have a huge amount of traffic, it will show you the routing decisions being made on a per-packet basis. You can lessen the load on your router by creating an access list that discards information about your internal network and then apply that access list to the debug command. Do a google search for "debug ip packet" for details. It's pretty simple. If you filter on information about your internal networks, all that will show up are packets being routed to external destinations.
4. Consider the applications and devices in use at this site. Do any of them access the Internet directly? Does that router have a complete view of your internal network, or is there a chance that it is missing some routes? If you're using a dynamic routing protocol, it should know about your network. If you're using static routes, verify that it has all the routes it needs
5. Here's one that may or may not work. I've never tried it. Create a loopback interface then change your default route to point toward your loopback interface. I suspect that if your traffic counters increase on that interface then that means that the default route is being used. You may even be able to turn on ip accounting or netflow on the loopback address to get more information. I've never tried this so I can't verify that it will work. It would be an interesting experiment, though.