In the field of Internet service evolution, classical and quantum paradigms are often seen as distinct and separate research areas. Typically, the classical Internet is perceived as a foundational communications infrastructure for the emerging quantum Internet, mainly providing essential services such as synchronization, signaling, control, and feedback. Current quantum Internet research requires a radical transformation of physical layer devices, often leading to the obsolescence of existing classical infrastructures. To address this challenge, in this work we introduce a new hybrid approach for the future development of the quantum Internet. Specifically, we will delve into various mechanisms and technologies that facilitate the integration of quantum networks with future 6G networks. Our goal is to develop a complete model for a hybrid quantum-classical Internet designed for future generation 6G in order to jointly and efficiently support both quantum and classical communications. In particular, we explore an innovative hybrid datalink layer architecture that is critical for effectively managing the diverse nature of quantum and classical data traffic and security. The main advantage of our model lies in the integration of new quantum hardware at the physical layer while leveraging existing advanced classical infrastructure. Our model is proposed as a step forward in the harmonization of quantum and classical Internet technologies, paving the way for a secure and fast future 6G architecture.