One of the most celebrated open problems in the theory of operator algebras is the Connes embedding conjecture, which (roughly speaking) asserts that every n-tuple of self-adjoint operators in a tracial von Neumann algebra can be approximated in distribution by n-tuples of finite-dimensional complex matrices. In this talk, we will discuss the problem of verifying this conjecture for a broad class of tracial von Neumann algebras, namely those arising from compact quantum groups. Within this framework, we show that the Connes embedding conjecture can be studied in terms of certain natural linear algebra questions related to quantum subgroups and their finite-dimensional representations. Using this approach, we establish that the von Neumann algebras associated to the free orthogonal and free unitary quantum groups have the Connes embedding property. We will also discuss some applications to free entropy. This talk is based on joint work with Benoit Collins and Roland Vergnioux.