Consider a sequence of independent q-Bernoulli trials with odds of success geometrically decreasing with rate q, 0<q<1. In this work, we introduce the process generated by making a step to the right when the outcome of the ith q-Bernoulli trial is “success” and a step to the left otherwise, for i=1,2,…,n. Furhtermore, the ith q-Bernoulli trial, happens during the ith part of a suitable defined partition of the time period (0,t], i=1,2,…,n. The position Xn,q(t) at time t of this process defines a q-random walk. Also, asymptotically, as n→∞, by a q-analogue of the De Moivre-Laplace theorem, we approximate this q-random walk by a q-Brownian motion. This q-Brownian motion is the continuous analogue of the q-random walk process and is distributed according to a linear transformed standardized Stieltjes-Wigert distribution.