This article envisions the concept of a ``Quantum Internet in the Sky", aiming to establish ubiquitous quantum communication links among distant nodes via free-space optical channels. Our key focus is on deploying quantum communication terminals on non-terrestrial platforms, specifically unmanned aerial vehicles and satellites, at various altitudes. By highlighting the unique characteristics of these platforms compared to terrestrial counterparts, we address inherent challenges and discuss potential solutions through meticulous system designs and analyses of typical non-terrestrial quantum communication scenarios. Finally, we illuminate the path forward by proposing essential future directions that underscore the integration of high-dimensional multipartite quantum communications with sensing, computing, and intelligence for multiple users en route to realizing a fully operational Quantum Internet.