New $q$-analogues of a congruence of Sun and Tauraso

Recently, by making use of a formula of Carlitz, the author confirmed a $q$-congruence conjectured by Tauraso. In this note, we use Carlitz's formula again to establish two new $q$-analogues of a congruence of Sun and Tauraso. We also give another generalization of Sun and Tauraso's congruence, along with its $q$-analogue.