Non-binary forward error correction (FEC) codes have been getting more attention lately in the coding society thanks mainly to their improved error correcting capabilities. Indeed, they reveal their full potential in the case of a oneto-one mapping between the code symbols over Galois fields (GF) and constellation points of the same order. Previously, we proposed non-binary FEC code designs targeting a given classical constellation through the optimization of the minimum Euclidean distance between candidate codewords. To go a step further, a better Euclidean distance spectrum can be achieved through the joint optimization of code parameters and positions of constellation symbols. However, this joint optimization for high order GFs reveals to be intractable in number of cases to evaluate. Therefore in this work, we propose a solution based on the multi-agent Deep Q-Network (DQN) algorithm. Applied to non-binary turbo codes (NB-TCs) over GF(64), the proposal largely improves performance by significantly lowering the error floor region of the resulting coded modulation scheme.