Not all middle-office risk functions are quantitative in nature though..and even then these tend to be filled mainly by ppl with PhDs or Masters in quantitative fields, though sometimes they may take in fresh grads with a strong quant background. This would include areas such as counterparty credit risk modelling & economic capital modellingetc. Stress testing is somewhat less quant, though some form of econometric modelling is usually required (depending on the approach the bank adopts).

Many of the roles however, in the context of a commercial bank, would be management or regulatory reporting, portfolio monitoring, policy, credit risk systems etc. which do not require a quant background. Often for these roles, banks are more open to hiring fresh grads. A possibility is to get hired into one of these roles and try for a transfer to a more quant role down the line, though I think some of these other more qualitative roles could be interesting in their own right...