A necessary condition called $H_\mu^{1,p}$-quasiconvexity on $p$-coercive integrands is introduced for the lower semicontinuity with respect to the strong convergence of $L^p_\mu(X;{\mathbb R}^m)$ of integral functionals defined on Cheeger-Sobolev spaces. Under polynomial growth conditions it turns out that this condition is necessary and sufficient.