In Australia, our House of Commons is called the House of Representatives, and we have a bicameral system like you do, with a Senate.
To the extent that the government of the day is made up of a majority of members of the House of Commons or House of Representatives—I know you have a minority government at the moment—it's that government that makes the recommendation to the Governor General on the appointment. It's fair to say that the members who comprise the government have a say in who's appointed, through the government.