I don't have one of the consoles but I have some experience trying to use those Serial to USB adapters, which would be the only place I would imagine an issue would come in since you already have it working on one computer. The HP versus Lenovo should not cause an issue. The thing that could cause an issue is the OS. Does the Thinkpad have the same OS as the HP? If so I would think it should work fine. The adapter I had, had issues in Linux and would only work at certain speeds (not all the speeds the adapter was supposed to support.) The adapter work fine in Windows Vista at all the speeds it said it would but the manufacture supplied drivers for that OS but not for Linux.
If you didn't have it working already on a computer I'd say to make sure the adapter supports the speed the console runs at and then make sure the device has proper drivers for the OS you are using so that you know it will actually work at that speed.