In this paper we define a neighbourhood semantics for quantified modal relevant logics. These models employ the Mares-Goldblatt [16] interpretation of the quantifiers, and are the natural combination of the semantics found in [27] and [9]. Modular soundness and completeness proofs are given in combination with an procedure for producing semantic conditions from axiom and rule schemes. We further show a model-wise equivalence between fully augmented neighbourhood models and relational models. Finally, we employ the constructed semantics to obtain an independence result surrounding the Barcan formulas, and their relation to various Augmentation principles.