There are no "qualifications" for QA Engineers in a 145 MRO.
It is "advised" that they should preferably be qualified to a similar standard of the Fleet being audited (eg Twin-engined Jets can be used for Bizjets to large Pax Jets) Though many QA staff are Licenced, they don't really need a Licence to do auditing!
Only if the QA staff are signatories for C of A's or ARC's should they have a Licence, with a 'similar type', and need to hold a Form 4 (and/or CAA Form 458).
Happybikers comment re SMS is good, but may not be suitable for a new QA engineer in a sizeable QA dept.
QA Managers shall be acceptable to the authority to hold a Form 4 (as stated in Pt145) and a CAA Form 458 if needed, and the authority may ask that the Form 4 holder be trained in some subjects such as QA, Maintenance Programmes and Airworthiness techniques as a condition of that acceptance ("Small" MROs should note that one!)
Many QA courses are usually advertised in Flight Infernal - but I don't know of their value personally (I did BSI's ISO9001 courses)