The coloured Jones and Alexander polynomials are quantum invariants that come from representation theory. There are important open problems in quantum topology regarding their geometric information. Our goal is to describe these invariants from a topological viewpoint, as intersections between submanifolds in configuration spaces. We show that the Nth coloured Jones and Alexander polynomials of a knot can be read off from Lagrangian intersections in a fixed configuration space. At the asymptotic level, we geometrically construct a universal ADO invariant for links as a limit of invariants given by intersections in configuration spaces. The parallel question of providing an invariant unifying the coloured Jones invariants is the subject of the universal Habiro invariant for knots. The universal ADO invariant that we construct recovers all of the coloured Alexander invariants (in particular, the Alexander polynomial in the first term).