Summary: Let \(H^{\infty}(\Omega,X)\) be the space of bounded analytic functions \(f(z)=\sum_{n=0}^{\infty} x_n z^n\) from a proper simply connected domain \(\Omega\) containing the unit disk \(\mathbb{D}:=\{z \in \mathbb{C}:|z| < 1\}\) into a complex Banach space \(X\) with \(\|f\|_{H^{\infty}(\Omega,X)} \leq 1\). Let \(\phi=\{\phi_n(r)\}_{n=0}^{\infty}\) with \(\phi_0 (r) \leq 1\) such that \(\sum_{n=0}^{\infty} \phi_n(r)\) converges locally uniformly with respect to \(r \in [0,1)\). For \(1\leq p\), \(q < \infty\), we denote \[ R_{p, q, \phi}(f, \Omega, X)= \sup \left\{r \geq 0: \|x_0\|^p \phi_0(r) + \left(\sum_{n=1}^{\infty} \|x_n\| \phi_n(r)\right)^q \leq \phi_0(r)\right\} \] and define the Bohr radius associated with \(\phi\) by \[ R_{p,q,\phi}(\Omega,X)=\inf \left\{R_{p,q,\phi}(f,\Omega,X): \|f\|_{H^{\infty}(\Omega,X)} \leq 1\right\}. \] In this article, we extensively study the Bohr radius \(R_{p,q,\phi}(\Omega,X)\), when \(X\) is an arbitrary Banach space, and \(X=\mathcal{B}(\mathcal{H})\) is the algebra of all bounded linear operators on a complex Hilbert space \(\mathcal{H}\). Furthermore, we establish the Bohr inequality for the operator-valued Cesàro operator and Bernardi operator.