Quaternion Kähler manifolds are known to be the target spaces for matter hypermultiplets coupled to N = 2 supergravity. It is also known that there is a one-to-one correspondence between 4n-dimensional quaternion Kähler manifolds and those 4(n+1)-dimensional hyperkähler spaces which are the target spaces for rigid superconformal hypermultiplets (such spaces are called hyperkähler cones). In this paper we present a projective-superspace construction to generate a hyperkähler cone M4(n+1)H of dimension 4(n+1) from a 2n-dimensional real analytic Kähler-Hodge manifold M2nK. The latter emerges as a maximal Kähler submanifold of the 4n-dimensional quaternion Kähler space M4nQ such that its Swann bundle coincides with N4(n+1)H. Our approach should be useful for the explicit construction of new quaternion Kähler metrics. The results obtained are also of interest, e.g., in the context of supergravity reduction N= 2 → N= 1, or alternatively from the point of view of embedding N= 1 matter-coupled supergravity into an N= 2 theory.