The Quantum Internet (QI) is a hypothetical global network infrastructure that exploits quantum mechanics principles to distribute qubits worldwide. The QI will enable breakthrough applications like unconditional security, distributed quantum computing, and quantum metrology. Nonetheless, the QI is currently a research-stage technology with daunting challenges until fully-functional quantum networks (QNs) can be realized. QNs are not standalone but require classical networks to assist the qubits exchange between remote places through quantum teleportation. On the other hand, QNs control and management planes will remain classical due to quantum networking technology manufacturing complexity and cost. This article addresses integrating QNs with Sixth Generation and Beyond (6GB) mobile networks. 6GB are the foreseen next generations of wireless networks, offering extensive ubiquity and improved ultra-reliable low-latency support over larger distances than their predecessors. Therefore, it is the best-suited candidate to enable potential mobile and non-terrestrial quantum applications. Unlike existing literature highlighting quantum technology’s potential benefits to mobile networks, this work presents a fresh vision in which 6GB enhance the QI and even contributes to accelerating its development. To that end, we identify target QNs scenarios enabled, facilitated, or benefited by 6GB. We propose a Software-Defined programmable 6GB-integrated QN architecture with slicing support and the associated procedure to create virtual QNs or slices. Last, we present a proof-of-principle showing that QNs can impose deterministic low-latency requirements out of 5G reach.