can I use the Arduino to check if a computer is sending data(to print) to any connected printer?
Conceivably you could insert the Arduino between the host and the printer and know whether there was communication between them, but understanding what the communication represented might need a lot of research.
I'm not sure what you're trying to achieve here, but if you want to look at the state of the printer and print queue then doing this from an application on the host PC seems like a far more sensible approach.