No. Qantas Frequent Flyer membership is provided on an individual basis.
You can only earn Qantas Points in your name. Each family member will need to have their own membership to earn Qantas Points.
Qantas Frequent Flyer membership is subject to the Terms and Conditions of the Qantas Frequent Flyer Program which can be viewed here.
A joining fee usually applies. However, this will be waived. Membership is subject to the Qantas Frequent Flyer Terms and Conditions and each application is subject to approval by Qantas.