Hi,
The HP Color LaserJet Pro MFP M479fdn is listed as supported in the article About AirPrint - Apple Support. This means that you should not have to install additional drivers, provided that this AirPrint printer and the new Mac are connected to the same network (Wi-Fi router). An AirPrint-enabled printer is compatible with macOS.
Since this printer is equipped with an Ethernet interface, a wired AirPrint network connection via one of the LAN ports of a router would be possible.
Add a printer to your printer list so you can use it on Mac – Apple Support (UK)
Software from the printer manufacturer may or may not add functionality.