In minimal leptoquark (LQ) models, the $R_2$ and $S_1$ can be the solution to the $(g-2)_{\mu}$ anomaly because of the chiral enhancements. Here, we study the LQ and vectorlike quark (VLQ) extended models. In the one LQ and one VLQ extended models, the $(g-2)_{\mu}$ can receive the contributions from top and top partner $T$ because of the $t-T$ mixing. Besides the traditional $R_2$ and $S_1$ representations, we find that the $S_3$ LQ can also explain the anomaly when including the $(X,T,B)_{L,R}$ triplet at the same time. Moreover, we find that the LQ has the new decay channel $T\mu$ in these models.