Let \(k\) be a separably closed field of arbitrary characteristic and let \(C\) denote a smooth projective curve over \(k\) having a \(k\)-rational point. For any scheme \(X\) over \(k\), let \(\mathrm{Br}'(X)\) denote the cohomological Brauer group of \(X\), which is, by definition, the torsion part of the étale cohomology group \(H^2_{\mathrm{et}}(X,\mathbb{G}_m)\). The present paper focuses on the way how the cohomological Brauer groups of \(\mathrm{Sym}^d(C)\), \(\mathrm{Pic}^d(C)\), and \(\mathrm{Quot}(r,d)\) are related to each other. Here \(\mathrm{Sym}^d(C)\) denotes the \(d\)-fold symmetric power of \(C\), \(\mathrm{Pic}^d(C)\) the Picard scheme of line bundles of degree \(d\) on \(C\), and \(\mathrm{Quot}(r,d)\) the quot-scheme parametrizing degree \(d\) quotients of \(\mathcal{O}_C^{\oplus r}\). Identifying the points of \(\mathrm{Sym}^d(C)\) with effective divisors of degree \(d\) on \(C\), one can define the Abel-Jacobi map \(\xi_d:\mathrm{Sym}^d(C)\rightarrow\mathrm{Pic}^d(C)\) by the map \(D\mapsto \mathcal{O}_C(D)\). On the other hand, given a point \(Q\in Q(r,d)\), one has the short exact sequence: \(0\rightarrow\mathcal{F}(Q)\rightarrow \mathcal{O}_c^{\oplus r}\rightarrow Q\rightarrow 0\). Sending \(Q\) to the scheme-theoretic support of the quotient for the induced homomorphism \(\bigwedge^r\mathcal{F}(Q)\rightarrow\bigwedge^r(\mathcal{O}_c^{\oplus r})\), one defines a morphism \(\phi_d:Q(r,d)\rightarrow \mathrm{Sym}^d(C)\). The main result of the paper shows the following: For any prime \(\ell\) invertible in \(k\), the induced maps \(\phi_d^*:\mathrm{Br}'(\mathrm{Sym}^d(C))_{\ell^n}\rightarrow \mathrm{Br}'(Q(r,d))_{\ell^n}\) and \(\xi_d^*:\mathrm{Br}'(\mathrm{Pic}^d(C))_{\ell^n}\rightarrow \mathrm{Br}'(\mathrm{Sym}^d(C))_{\ell^n}\) are isomorphisms for any \(n>0\) and for any \(d\geq 3\). Over the field of complex numbers, it is shown in [I. Biswas et al., Mich. Math. J. 64, No. 3, 493–508 (2015; Zbl 1327.14097)] that both maps \(\phi_d^*\) and \(\xi_d^*\) are isomorphisms for all \(d\geq 2\). In order to generalize to the positive characteristic case, the authors employ the result in [A. Collino, Ill. J. Math. 19, 567–583 (1975; Zbl 0346.14003)], which calculates the Chow groups of symmetric powers of curves, and the calculation in [S. del Baño, J. Reine Angew. Math. 532, 105–132 (2001; Zbl 1044.14005)] of Picard groups of the Quot-schemes. They also obtain similar results for Prym varieties.