In a word, No. "Q" registrations are given to vehicles as replacement marks as the age is difficult to determine. Usually "Q" plates are given to kit-cars where the vehicle is made up of various other vehicles.
Can I Sell My Q Plate
At this point in time, "Q" plates are non-transferrable and so they cannot be sold either. The DVLA may issue an age-related vehicle registration number if sufficient documentary evidence is submitted to prove a vehicle's age but at present transfers to "Q" plated vehicles are banned.