Both the modules have the same operating frequency i.e. 125 KHz and support Serial RS232/TTL output while the operating frequency for RC522 is 13.56 kHz. Circuit Diagram for Interfacing RDM6300 with ...

You can try to convert the TTL-Signal directly by Software to RS232 but be reminded, that pure RS232 is 12V while the arduino uses 5V for signal high values. There is a commented line in the example ...