<p>The central aim of the current paper is to delve into the radii of starlikeness and convexity for two distinct normalizations of the big <i>q</i>-Bessel function, both of which are analytic within the confines of the unit disk of the complex plane. Additionally, by demonstrating that both functions possess an infinite number of zeros, each of which is a real number, the infinite product representations for these functions are derived. It is crucial to highlight that the Laguerre-Pólya class of real entire functions, the Hadamard factorization theorem, and the Euler-Rayleigh inequalities are key components of this study. Finally, by applying the Euler-Rayleigh inequalities to the least positive zeros of the normalized big <i>q</i>-Bessel functions, tight bounds, both lower and upper, for the radii of starlikeness and convexity are derived in this study.</p>