  Let $\mathbb{X}_{\boldsymbol{p},\boldsymbol{\lambda}}$ be a weighted
projective line. We define the quantum cluster algebra of
$\mathbb{X}_{\boldsymbol{p},\boldsymbol{\lambda}}$ and realize its specialized
version as the subquotient of the Hall algebra of
$\mathbb{X}_{\boldsymbol{p},\boldsymbol{\lambda}}$ via the quantum cluster
character map. Inspired by \cite{Chen2021}, we prove an analogue cluster
multiplication formula between quantum cluster characters. As an application,
we obtain the polynomial property of the cardinalities of Grassmannian
varieties of exceptional coherent sheaves on
$\mathbb{X}_{\boldsymbol{p},\boldsymbol{\lambda}}$ . In the end, we construct
several bar-invariant $\mathbb{Z}[\nu^{\pm}]$-bases for the quantum cluster
algebra of the projective line $\mathbb{P}^1$ and show how it coincides with
the quantum cluster algebra of the Kronecker quiver.
