We present a comprehensive theoretical investigation of high-order harmonic generation in H$^+_2$ molecules within a quantum-optical framework. Our study focuses on characterizing various quantum-optical and quantum-information measures stemming from the correlations established between light and matter. We demonstrate the emergence of entanglement between electron and light states after the laser-matter interaction. We also identify the possibility of obtaining nonclassical states of light in targeted frequency modes by conditioning on specific electronic quantum states, which turn out to be crucial in the generation of highly nonclassical entangled states between distinct sets of harmonic modes. Our findings open up avenues for studying strong-laser-field-driven interactions in molecular systems under a fully quantum-mechanical framework.