  We explore how the quantum entanglement is modified in the generalized
uncertainty principle (GUP)-corrected quantum mechanics by introducing the
coupled harmonic oscillator system. Constructing the ground state $\rho_0$ and
its reduced substate $\rho_A = \mbox{Tr}_B \rho_0$, we compute two entanglement
measures of $\rho_0$, i.e. ${\cal E}_{EoF} (\rho_0) = S_{von} (\rho_A)$ and
${\cal E}_{\gamma} (\rho_0) = S_{\gamma} (\rho_A)$, where $S_{von}$ and
$S_{\gamma}$ are the von Neumann and R\'{e}nyi entropies, up to the first order
of the GUP parameter $\alpha$. It is shown that ${\cal E}_{\gamma} (\rho_0)$
increases with increasing $\alpha$ when $\gamma = 2, 3, \cdots$. The remarkable
fact is that ${\cal E}_{EoF} (\rho_0)$ does not have first-order of $\alpha$.
Based on there results we conjecture that ${\cal E}_{\gamma} (\rho_0)$
increases or decreases with increasing $\alpha$ when $\gamma > 1$ or $\gamma <
1$ respectively for nonnegative real $\gamma$.
