A $q$-analogue of Gosper's strange evaluation of the hypergeometric series

In 1977, Gosper conjectured many strange evaluations of hypergeometric series. One of them is a ${}_2{F}_1$-series identity with two free parameters, which was proved by Ebisu (2013), Chu (2017), and Campbell (2023) in different ways. In this paper, we present a $q$-analogue of the ${}_2{F}_1$-series identity, along with its generalization, by using three-term relations for the ${}_2{\phi }_1$ basic hypergeometric series.