The requirements for the Fast Track F/O are at the time of joining.
If you join before obtaining the Fast Track F/O requirements then you will considered as a normal F/O and have to follow normal F/O upgrade procedure and requirements.
With your 3100 total turbo prop/jet time I do not think it would make a difference if you join QR now or after you obtain the requirements because by the time you get your hours from your employer or with QR you will have spent the same amount of time, about 3 years.