  Let $B, B'\subset \mathbb{R}^d$ with $d\geq 2$ be two balls such that
$B'\subset \subset B$ and the position of $B'$ is varied within $B$. For $p\in
(1, \infty ),$ $s\in (0,1)$, and $q \in [1, p^*_s)$ with
$p^*_s=\frac{dp}{d-sp}$ if $sp < d$ and $p^*_s=\infty $ if $sp \geq d$, let
$\lambda ^s_{p,q}(B\setminus \overline{B'})$ be the first $q$-eigenvalue of the
fractional $p$-Laplace operator $(-\Delta _p)^s$ in $B\setminus \overline{B'}$
with the homogeneous nonlocal Dirichlet boundary conditions. We prove that
$\lambda ^s_{p,q}(B\setminus \overline{B'})$ strictly decreases as the inner
ball $B'$ moves towards the outer boundary $\partial B$. To obtain this strict
monotonicity, we establish a strict Faber-Krahn type inequality for $\lambda
_{p,q}^s(\cdot )$ under polarization. This extends some monotonicity results
obtained by Djitte-Fall-Weth (Calc. Var. Partial Differential Equations,
60:231, 2021) in the case of $(-\Delta )^s$ and $q=1, 2$ to $(-\Delta _p)^s$
and $q\in [1, p^*_s).$ Additionally, we provide the strict monotonicity results
for the general domains that are difference of Steiner symmetric or foliated
Schwarz symmetric sets in $\mathbb{R}^d$.
