The paper extends the \(q\)-entropy theory from symbolic to general topological dynamical systems. Specifically, by means of a weak Gibbs measure, the authors define the \(q\)-topological entropy and the \(q\)-metric entropy, and then study their basic properties. In particular, they describe the relationships between \(q\)-topological entropy and topological pressure for almost additive potentials, and between \(q\)-metric entropy and local metric entropy.