Does Qantas have an above-average safety record?
Defining and Measuring 'Safety'

These ranking depends how you define safety. Here is your problem: You can easily do this by widely publicised facts, such as well documented fatal accidents. Fatal accidents occur much much too seldom to with any statistical confidence to say how safe an airline is. 

What you would much rather do is go by accidents and incidents (the little things that occur everyday and give you a feeling for how stuff is done and handled). For instance: hydraulic leaks, loss of separation to other aircraft or electrical problems prompting a return. A small selection of these are reported on the popular site Aviation Herald, but this is (emphasis) only a small subset that happen to come through. Some western countries with the objectively the best safety are over-reported since they are just better at documenting. You'll find heaps of entries on 'fume events' on Airbus aircraft from Germany for instance.

These are considerably harder to collect, since how they are classified and reported changes by country and aviation agency. Some publish for common good, other ones barely collect them. For this reason, the way these 'rankings' do it is by fatal accidents and other easy to get parameters. This is eventually what comes through to newspaper sites with little detail on how they were put together.

The parameters measured by AirlineReports:


Is the airline IOSA certified?
Is the airline on the European Union (EU) Blacklist?
Has the airline maintained a fatality free record for the past 10 years? 
Is the airline FAA endorsed?
Does the country of airline origin meet all 8 ICAO safety parameters?
Has the airline's fleet been grounded by the country's governing aviation safety authority due to safety concerns?
Does the airline operate only Russian built aircraft?


Most of these are bare minimums that can be expected from an airline, but do not really go to any distance to show who stands out. For any more detailed sense and purposes (how media are using it), I'd call this useless. 

Also note that this does not include the liability question for fatal accidents- who was at fault? You can finally also (albeit unlikely) have a completely safe airline that does not have a single certification.

Qantas

Qantas has had two very serious incidents with a Boeing 747 and A380. They also had a serious incident on involving A330 losing altitude. This does not really IMHO award them as the 'safest airline' for the past few years, since this is considerably more than a lot of other airlines have experienced. 

However, for all practical sense and purposes, they are as safe as any other airline. The chances of anything happening are just so incredibly remote that it's not worth the trouble of thinking about it. Getting a connecting flight to fly with another airline will probably be less safe than just flying them directly.

As for the maintenance matters, Qantas owns JetStar and they appear to use the same maintenance facility for their aircraft. A further difference is the training where there appear to be overlaps as well, so there are only small differences on perhaps the most important parameters differing the two.