We define q-analogues of Fourier-Bessel series, by means of complete q-
orthogonal systems constructed with the third Jackson q-Bessel function. Sufficient
conditions for pointwise convergence of these series are obtained, in terms of a
general convergence principle valid for other Fourier series on grids defined over
numerable sets. The results are illustrated with specific examples of developments
in q-Fourier-Bessel series