Let $R$ be the finite chain ring $\mathbb{F}_{p^{2m}}+{u}\mathbb{F}_{p^{2m}}$, where $\mathbb{F}_{p^{2m}}$ is the finite field with $p^{2m}$ elements, $p$ is a prime, $m$ is a non-negative integer and ${u}^{2}=0.$ In this paper, we firstly define a class of Gray maps, which changes the Hermitian self-orthogonal property of linear codes over $\mathbb{F}_{2^{2m}}+{u}\mathbb{F}_{2^{2m}}$ into the Hermitian self-orthogonal property of linear codes over $\mathbb{F}_{2^{2m}}$. Applying the Hermitian construction, a new class of $2^{m}$-ary quantum codes are obtained from Hermitian constacyclic self-orthogonal codes over $\mathbb{F}_{2^{2m}}+{u}\mathbb{F}_{2^{2m}}.$ We secondly define another class of maps, which changes the Hermitian self-orthogonal property of linear codes over $R$ into the trace self-orthogonal property of linear codes over $\mathbb{F}_{p^{2m}}$. Using the Symplectic construction, a new class of $p^{m}$-ary quantum codes are obtained from Hermitian constacyclic self-orthogonal codes over $R.$