They can communicate with the proper adapter, but probably not the way you are envisioning. As mentioned in the comments, the RJ45 connector on your cable is not connected the same as an Ethernet RJ45 connector. This would be considered a null-modem connection. You will need the appropriate DB9-to-RJ45 modular connector to get standard connections on both ends. Next, you will probably need some USB to RS-232 adapters unless your computer still has built-in COM ports. Even still, you will only be able to communicate using RS-232 serial communications. You would need some sort of terminal emulator software on both computers.
IP-based programs actually can communicate over these cables using the good old 1988 technology of SLIP or its replacement, PPP. This used to be a fairly common way to provide networking to simpler equipment, back when NICs were expensive.
Due to the limits of RS-232 serial connections, be aware that your maximum throughput will literally be about 10,000 times slower than a gigabit Ethernet crossover cable.
So, it can be done, but it probably shouldn't be done.