Identifying the unknown convolutional code corresponding to the given intercepted data is an important problem in military surveillance and in wireless communication. While a variety of code identification algorithms are available in the literature, the key contribution of our work lies in the novel solution and the corresponding analysis. In this paper, we focus on the situation when the given data corresponds to either of the two potential convolutional codes and the goal is to detect the correct code. We first provide a new interpretation of the convolutional code as a Markov chain, which is more suitable for analyzing the code detection problem. Our problem then gets reduced to identifying between the two Markov chains. We provide the closed-form expressions for the corresponding state transition matrices and estimate the error exponent for the underlying likelihood ratio test (LRT). We also provide a computationally efficient BCJR-based method for computing the likelihoods required for the LRT. We observe that BCJR-based likelihoods suffer from numerical issues for a longer data sequence, and hence, in this case, we design neural networks that have been found to achieve the optimal performance of the LRT.