There are a number of ways you can do the trip. Which is the best way? That is up to you. Do you want to go to ATH only?
A return award PER-ATH is 128,000 points + taxes and charges. One possible route is PER(QF71)-SIN(QF31)-LHR(BA640)-ATH(BA631)-LHR(QF32)-SIN(QF72)-PER. As I mentioned before you can route via HKG or even NRT.
If you want to visit other places, and have the time to do it, then why not redeem a OneWorld RTW award for 140,000 points (142,500 points if requiring phone assistance) + taxes & charges, including up to 5 stopovers and a limit of 35,000 flown miles. An example itinerary could be PER-SIN-LHR-ATH-MAD-JFK-LAX-HNL-SYD-PER.
QF FF awards are not available using codeshare flight numbers with the specific exception of LA operate flights on SYD-AKL-SCL and return. All other QF FF awards must be on the operating airline's flight number. And since GF is not a QF FF partner, you cannot use QF FF points to redeem awards on GF operated services.
So you will need to route via LHR using BA for LHR-BAH-LHR and the again use BA for LHR-ATH-LHR. It will probably be most cost-effective to use the OneWorld award and include a CX or IB flight in the mix to ensure you have two carriers other than QF. That award is based on the total miles flown (as opposed to one-way trips for other QF awards) and includes up to 5 stopovers.
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