Let \(B, B'\subset \mathbb {R}^d\) with \(d\ge 2\) 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 \(sp \ge d\), 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 \(\mathbb {R}^d\).