<p>The q-calculus, or quantum calculus, is a relatively new branch of mathematics that allows for the calculation of derivatives of real functions without using limit. In this paper, we focus on the logistic initial value problem using the newly defined q-derivatives in fractional quantum calculus. By utilizing the properties of the q-gamma function and applying the fractional q-integral, we were able to derive an equivalent q-integral equation corresponding to the given Caputo quantum initial value problem of the logistic equation. Our emphasis is on proving the existence and uniqueness of the solution. To demonstrate the uniqueness of the solution, we employ the Banach fixed point theorem, and to prove the existence of a solution, we use the Brouwer fixed point theorem. Additionally, we utilize a numerical method to simulate the solution of the q-fractional logistic equation under different parameters.</p>