If you have not already credited your past flights to any program, you may be in luck, as according to the Qatar Airways FAQ:
How can I claim my missing Qmiles with partners? Only missing Qmiles requests for airline partner transactions will be accepted by Privilege Club. Please write to [email protected] with copies of your ticket and boarding pass. Claims can be made up to 30 days prior to the date of joining Privilege Club for new members and within 180 days after the date of travel for existing members.
So if your BA flight was within the last 30 days, you can join the QR Privilege Club program and request credit for it. You can use the "QCalculator" to determine how many miles or points you would be entitled to, which will be based on the fare class you booked into. However, airlines give miles for partner flights begrudgingly, so it is possible they will decline your request, especially if you tend to fly discount economy. Some investment of time with customer service may be required.
If you have already credited the flight as BA Avios or to another program, you are out of luck; "double-dipping" is prohibited by most programs, including Qatar's— after all, they are trying to promote exclusive loyalty. The Privilege Club Terms and Conditions explicitly state
7.4 You cannot earn Qpoints if the flight is already credited in another loyalty programme. Qatar Airways reserves the right to deduct any Qpoints if the same flight is credited for the purposes of tier status accrual or with miles, points or any other form of redeemable credits/mileage on any other loyalty programme.
8.6 Qatar Airways and any Partner may place restrictions on your mileage accrual. You cannot earn Qmiles if the flight or Transaction is already credited in another loyalty programme. Qatar Airways reserves the right to deduct any Qmiles if the same flight or Transaction is credited with miles, points or any other form of redeemable credits/mileage on any other loyalty programme.
Moreover, BA is a dead-end program; there are currently no partners through which you can transfer miles out. Indeed, it is almost impossible to transfer miles out of major airline programs now, and as calculators like the WebFlyer.com Mileage Converter demonstrate, the few existing options are at truly ruinous exchange rates.