  We propose the refined topological string correspondence to the expectation
values of half-BPS Wilson loop operators in 5d $\mathcal{N}=1$ gauge theory
partition function on the Omega-deformed background
$\mathbb{R}^4_{\epsilon_{1,2}}\times S^1$. We provide the refined topological
vertex method and the refined holomorphic anomaly equation method in the
topological string theory, from which we have exact computations on the 5d
Wilson loops partition functions in both A- and B-models. Finally, with the
exact results we have in B-model, we recover the quantum periods of local
$\mathbb{P}^1\times\mathbb{P}^1$ model and local $\mathbb{P}^2$ model in the
study of quantum geometry and we further give a refined generalization of
A-period.
