Can someone please tell me if I can use the nRF24L01 on the arduino NANO?

Yes, you don’t have to ask permission

Yes you can and Robin2's simple rf24 tutorial will help you do so.

Wireless problems can be very difficult to debug so get the wireless part working on its own before you start adding any other features.

The examples are as simple as I could make them and they have worked for other Forum members. If you get stuck it will be easier to help with code that I am familiar with. Start by getting the first example to work

There is also a connection test program to check that the Arduino can talk to the nRF24 it is connected to.

A common problem with nRF24 modules is insufficient 3.3v current from the Arduino 3.3v pin. This seems to e a particular problem with the nano. The high-power nRF24s (with the external antenna) will definitely need an external power supply. At least for testing try powering the nRF24 with a pair of AA alkaline cells (3v) with the battery GND connected to the Arduino GND.

If you are using the high-power nRF24s (with the external antenna) make sure there is sufficient distance between the two nRF24s so that the signal does not overwhelm the receiver - try 3 metres separation. If you are new to nRF24s it may be better to start with a pair of low power modules with the pcb antenna.