Your HP Photosmart 7510 ought to work with AirPrint, so the only uncertainty is your router. It must support Bonjour for AirPrint to work. All Apple AirPort base stations do (obviously) but most others should as long as they are fairly new. They may require a firmware update for AirPrint to function though.
I am not familiar with yours and its website yields almost no information about AirPrint compatibility.
There is firmware available for it, but I have no idea how it is uploaded to the router. See this:
http://www.zyxel.com/support/download_library/product/nbg_419n_7.shtml
ZyXEL also has a user support forum here: http://forum.zyxel.com
Perhaps you could find someone more knowledgeable about it on that site.