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Entanglement-assisted quantum error-correcting codes, which can be seen as a generalization of quantum error-correcting codes, can be constructed from arbitrary classical linear codes by relaxing the self-orthogonality properties and using pre-shared entangled states between the sender and the receiver, and can also improve the performance of quantum error-correcting codes. In this paper, we construct some families of entanglement-assisted quantum maximum-distance-separable codes with parameters $[[\frac{{{q^2} - 1}}{a},\frac{{{q^2} - 1}}{a} - 2d+2 + c,d;c]]_q$, where $q$ is a prime power with the form $q = am \pm \ell$, $a = \frac{{\ell^2} - 1}{3}$ is an odd integer, $\ell \equiv 2\ (\bmod\ 6)$ or $\ell \equiv 4\ (\bmod\ 6)$, and $m$ is a positive integer. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature.

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