Well, no, neither can I. Reading through the discussion at Talk:Australia, it seems most are arguing for it's exclusion because the infobox is only to deal with the most basic information, showing only the national flag, the national anthem, the national arms, etc.  God Save the Queen may not be the national anthem, but it seems to me that it was decreed to be an anthem that applies nationally - it is the Royal Anthem all across Australia, and always so, not just when a member of the Royal Famliy is present. 
 ::Additionally, as for precedent, not all articles on countries that are monarchies list a royal anthem; whether this is because there actually isn't one I don't know.  Of the Commonwealth Realms, currently Australia, Barbados, and Saint Vincent and the Grenadines do not list a royal anthem in the infobox.  Non-Commonwealth Realm monarchies that do have a royal anthem in the infobox include Sweden, Norway, and perhaps Spain - the national anthem is Marcha Real, or Royal March. 
 :Perhaps this is something that should be brought up as a Wiki-wide policy. In fact, a look at Wikipedia:WikiProject Countries shows no provision even for the national anthem to be included in the infobox.  Maybe your concerns should be raised there.  I would follow along, of course.