  With the method of the background field expansion, we investigate the
one-loop quantization of the Euclidean nonlocal $f(R)$ model in the de-Sitter
universe. We obtain the ghost-free condition (GFC) based on the transformation
from the Jordan frame to the Einstein frame and the classical stability
condition (CSC) satisfied $f^{(0)}_{RR}-\phi_0F^{(0)}_{RR}<0$. We present the
on-shell and off-shell one-loop effective action and quantum stability
condition (QSC) by utilizing the generalized zeta-function. We find that under
the fulfillment of GFC, CSC and QSC are inconsistent.
