I've observed that there are, broadly, two routes into consulting. Mars knows the field as well as anyone and better than most.
Route 1 = join one of the large companies' graduate schemes straight from university, e.g. Deloitte, Accenture, PWC, McKinsey, Bain etc. As far as I can tell, the subject of your degree is not particularly important. Companies like to have a large amount of bright, eager, competitive grads to do the grunt work, recognising that a large proportion of people will move on, relatively quickly - up or out.
Route 2 = join a company as an experienced hire having picked up many of the skills valued by whatever consultancy work you will be doing. I had worked in the public sector for 10 years, so knew the culture of many departments and had several years experience of soft skills training and managing medium sized projects.
I don't know of any consultancy qualifications. As always, the important thing is to have the qualifications needed to get the job you want. Have you approached any companies already and if so what was the response?