While much of the current study on quantum computation employs low-level formalisms such as quantum circuits, several high-level programming languages/calculi have been recently proposed aiming at structured quantum programming. The current work contributes to the semantical study of such languages, by providing interaction-based semantics of a functional quantum programming language; the latter is based on linear lambda calculus and is equipped with features like the modality and recursion. The proposed denotational model is the first one that supports the full features of a quantum functional programming language; we also prove adequacy of our semantics. The construction of our model is by a series of existing techniques taken from the semantics of classical computation as well as from process theory. The most notable among them is Girard's Geometry of Interaction (GoI), categorically formulated by Abramsky, Haghverdi and Scott. The mathematical genericity of these technique--largely due to their categorical formulatio--is exploited for our move from classical to quantum.