The authors consider the \(q\)-Sturm-Liouville operator \[ l(y):=-\frac{1}{q}D_{q^{-1}}D_qy(x)+w(x)y(x),\quad 0\leq x\leq a<+\infty. \] In the above operator, it holds that \(w\) is defined on \([0,a]\) and is continuous at \(0\). Moreover, \(0<q<1\). The authors then consider \(L_0\), which is the closure of the minimal operator generated by \(l\) and then argue that the maximal operator, \(L\), is given by \(L=L_0^*\). The main result of the paper is that all eigenvalues of the operator \(L\) lie on the open upper half-plane and are purely discrete. Furthermore, a completeness result regarding the eigenfunctions of \(L\) in the space \(L_q^2(0,a)\) is established; note that \(L_q^2(0,a)\) is the space of all complex-valued functions \(f\) defined on \([0,a]\) such that \[ \| f\|:=\left(\int_0^a |f(x)|\;d_qx\right)^{\frac{1}{2}}<+\infty. \]