Magnons are quantized collective spin-wave excitations in magnetically ordered systems. Revealing their interactions among these collective modes is crucial for the understanding of fundamental many-body effects in such systems and the development of high-speed information transport and processing devices based on them. Nevertheless, identifying couplings between individual magnon modes remains a long-standing challenge. Here, we observe unambiguous spectroscopic fingerprints of anharmonic coupling between distinct magnon modes in an antiferromagnet, as evidenced by coherent photon emission at the sum and difference frequencies of the two modes. This discovery is enabled by driving two magnon modes coherently with a pair of tailored terahertz fields and then disentangling a mixture of nonlinear responses with different origins, symmetries, and field dependences in a two-dimensional frequency-frequency correlation map. Our approach provides a new platform for generating nonlinear magnonmagnon mixings and establishes a systematic means of unveiling intricate couplings among distinct low-energy collective modes.