Aq-analogue of a formula of G.N. Watson that gives the product of two non-terminating Gaussian hypergeometric functions as a sum of twoF4Appell series is obtained. of the many possibleq-extensions of the2F1function, the one that reduces to the well-known Askey-Wilson polynomial when specialized to a terminating series, is chosen. The product formula is shown to reduce to Watson's formula in the limitq→ 1.