In this article, we extend our Distributed Lagrange Multiplier/Fictitious Domain method previously implemented on simple regular Cartesian grids to quadtree/octree adaptive grids. The objective is to improve both the accuracy and efficiency of our DLM/FD particle-resolved simulation method by extending its computing capabitilies through dynamic local mesh refinement. The main features of our numerical method, such as a first-order operator splitting time algorithm and a second-order reconstruction of the velocity field close to the boundary of the immersed rigid bodies (of arbitrary shape), are unchanged. We implemented our adaptive DLM/FD algorithm within Basilisk, a parallel platform to solve partial differential equations on dynamic quadtree/octree grids. The quadtree/octree structure of the grid and specific design rules of Basilisk impose a special treatment of some of the operations performed on the grid in the DLM/FD-Uzawa algorithm. The new computational method is then tested and validated on a set of flow configurations including the challenging problem of accurately computing lubrication interaction forces without resorting to using any ad hoc correction. Finally, we illustrate the potential of our code to compute complex particle-laden flow configurations that were not attainable in the past with a DLM/FD algorithm implemented on a simple regular Cartesian grid.