This article is the second part of the work by the authors on the quantum \(j\)-invariant \(j^{\mathrm{qt}}\) in positive characteristic. The first part appears in the same volume [ibid. 107, No. 1, 23–35 (2016; Zbl 1343.11068)]. In this second part, the authors deal with values of the invariant \(j^{\mathrm{qt}}\) defined in the first part at quadratic elements of \(k_\infty\), where \(k_\infty\) is the completion of the function field \(\mathbb F(T)\) over a finite field \(\mathbb F\) with respect to a prime element \(1/T\). The invariant \(j^{\mathrm{qt}}\) is the analog of modular invariant for quantum tori introduced in [C. Castaño-Bernard and the second author, Proc. Lond. Math. Soc. (3) 109, No. 4, 1014–1049 (2014; Zbl 1360.11075)] and is a multi-valued modular function defined to be limit of a sequence. Let \(f\) be a quadratic element satisfying the equation \(X^2-aX-b=0\) with \(a\in \mathbb F[T]\backslash \mathbb F\) and \(b\in\mathbb F^\times\). Then the authors show that \(j^{\mathrm{qt}}(f)\) has just \(\deg a\) values and obtain these values in rational functions of \(f\) explicitly. Further, from this result they deduce that at any quadratic element \(h\) the number of values of \(j^{\mathrm{qt}}(h)\) is finite.