Well, maybe. First, as others have said, you would HAVE to ensure the wires you use are disconnected from everything else.
The "maybe" part is because of line length and losses. A typical Serial Port on a computer is actually a very specific port type, RS-232. It uses mainly voltage-type signals and high-impedance 5 VDC circuits for communication. Such circuits work well over shorter distances and in the absence of strong external sources of electrical noise. Hence they are well suited to connecting a computer to a printer that is withing 5 to 10 feet. But there is a real limit to how far that connection can extend and still be reliable. For longer distances, one older way was another version of serial data transfer called RS-422 that used low-impedance circuitry and a current-based signal, not voltage-based. It was much better at noise resistance and less likely to suffer from weakened signal over a long line. But that system is almost never found built into a computer mobo or a peripheral device.
There are lots of other ways to connect devices over longer distances, if that is your need. But most of them involve using or installing a different cable. Your question was how NOT to have to do that, and just use what's already there. So, MAYBE it could work.