The partition algebra is the centralizer of the symmetric group acting on tensor powers of its natural (permutation) module. It has a diagrammatic basis that generalizes Brauer's centralizer algebra for the orthogonal group. Many of these diagram centralizer algebras have q-generalizations: for example the q-symmetric group is the Iwahori-Hecke algebra and the q-Brauer algebra is the BMW algebra. We will introduce a candidate for a q-partition algebra --- constructed using Harish-Chandra restriction and induction on the finite general linear group over a field with q elements --- and we will illustrate some preliminary computations in this algebra.