Summary: In this paper, we study the following \((p, q)\)-Kirchhoff problem with Choquard nonlinearity: \begin{align*} - & (1+a \int\limits_{\mathbb{R}^N}|\nabla u|^pdx)\Delta_p u\\ - & (1+b\int\limits_{\mathbb{R}^N}|\nabla u|^qdx)\Delta_q u+V_{\varepsilon}(x)(|u|^{p-2}u+|u|^{q-2}u)\\ = & (|x|^{-\mu}\ast F(u))f(u)\text{ in }\mathbb{R}^N, \end{align*} where \(\varepsilon\) is a small positive parameter, \(a,b\) are positive constants, \(1<p<q<N\), \(q<2p\), \(\Delta_su=\operatorname{div}(|\nabla u|^{s-2}\nabla u)\) with \(s\in\{p,q\}\) is the \(s\)-Laplacian, the potential \(V:\mathbb{R}^N\to\mathbb{R}\) is continuous, \(V_{\varepsilon}(x)=V(\varepsilon x)\), \(0<\mu<q\), \(f\) is a continuous nonlinearity, and \(F\) is the primitive of \(f\). The main result in this paper establishes multiplicity and concentration properties of positive solutions under weaker hypotheses. The proofs combine nonstandard Nehari manifold methods, penalization techniques and the Ljusternik-Schnirelmann category theory.