Qantas IS a BA partner, every oneworld airline is a BA partner.
As a general point, just becasue a web form allows you to enter a number, does not mean that will earn anything - you need to check the rules of the programme you want to credit to.
In this case, you want to credit to BA Executive Club, so you check BAEC website, on the link I posted above:
http://www.britishairways.com/travel...d/public/en_gb
Which says:
"You can collect Avios points and earn Tier Points when you fly with any of our oneworldŽ alliance partners across an unrivalled international route network."
Then you look down the page at Qantas, and it tells you how many Avios and TP you will earn from different flights.
Check your flights on the Qantas website to see which booking class you are booked into. Compare with the tables on BA website. That gives you your answer.
Some fare codes will not give Avios and TP, you need to check the BA tables, e.g. if Qantas says you are booking into an 'E' fare, it does not earn in BAEC.
Assuming you are booked into a fare bucked listed, you will earn Avios and TP in BAEC.
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Same applies to any other airilne. Check what BAEC says you should earn in BAEC from fligths with that airline.