I have similarly two separate flights on Qatar booked (two separate awards, no place I could find (or had points) had both legs available to book as one) so we're flying FCO-DOH in J and DOH-SYD in F later this year, with approx 3h 40m between flights.
I found this on QR's website which led me to proceed without too much worry:
https://www.qatarairways.com/en/abou.../oneworld.html (scroll down and click "Through Check In" under "Other benefits")
It doesn't explicitly state that QR-QR is allowed, but Qatar is a oneworld airline, so it should be permitted, I'd think/hope!
I'd similarly love any feedback on others who have done this, though.