No. Computer software is limited to the Operating System of the Computer. Consoles do not run the same OS, and thus can't run the 'code' the same way.
Now the only way possible as your suggesting (which is a clusterfud) would be your PC has a TV Card to INPUT TV signal to 'see' on the computer screen, and you hooked the Console to that TV Card connection to 'send' the console view onto the Computer's "TV" window then run a record software to record that.