  We explore a possible time variation of the fine structure constant ($\alpha
\equiv e^2/\hbar c$) using the Sunyaev-Zel'dovich effect measurements of galaxy
clusters along with their X-ray observations. Specifically, the ratio of the
integrated Compto-ionization parameter $Y_{SZ}D_A^2$ and its X-ray counterpart
$Y_X$ is used as an observable to constrain the bounds on the variation of
$\alpha$. Considering the violation of cosmic distance duality relation, this
ratio depends on the fine structure constant as $\sim \alpha^3$. We use the
quintessence model to provide the origin of $\alpha$ time variation. In order
to give a robust test on $\alpha$ variation, two galaxy cluster samples, the 61
clusters provided by the Planck collaboration and the 58 clusters detected by
the South Pole Telescope, are collected for analysis. Their X-ray observations
are given by the XMM-Newton survey. Our results give
$\zeta=-0.203^{+0.101}_{-0.099}$ for the Planck sample and
$\zeta=-0.043^{+0.165}_{-0.148}$ for the SPT sample, indicating that $\alpha$
is constant with redshift within $3\sigma$ and $1\sigma$ for the two samples,
respectively.
