E. C. Titchmarsh proved some theorems (Theorems 84 and 85) on the classical Fourier transform of functions satisfying conditions related to the Cauchy-Lipschitz conditions in the one-dimensional case. In this paper, we obtain a generalization of those theorems for the $ q $-Bessel transform of a set of functions satisfying the $ q $-Bessel-Lipschitz condition of certain order in suitable weighted spaces $\mathcal{L}^{p}_{q,\nu}(\mathbb{R}_{q}^{+})$, where $1 < p \leq 2$. In addition, we introduce the $ q $-Bessel-Dini-Lipschitz condition and we obtain analogous of Titchmarsh’s theorems in this occurrence.