Congratulations!
A quick peek at the T&C's doesn't appear to specify any age limit for membership, but definitely excludes infant fares from eligibility to earn points (though it does indicate that if said infant is occupying their own paid seat, then miles can be earned).
9.4.1 Points are not earned for travel on:
(a) flights or booking classes that are not listed in or are excluded in the Airline Earning Table;
(b) infant fares (infant fares do not include infants travelling in their own paid seat);
(c) charter flights;
(d) freighter flights;
(e) free tickets or Award Flights;
(f) travel industry rebated tickets;
(g) competition prize tickets;
(h) transport other than air transport unless specified by Qantas;
(i) tickets acquired at an auction; or
(j) tickets which are subject to any specific provisions excluding the accumulation of Points.