Recent investigations of type Ia supernovae (SNeIa) suggest that the expansion of the Universe is accelerating, driven by some kind of negative-pressure dark energy. Independent evidence for the SNeIa results is provided by observations of cosmic microwave background (CMB) anisotropies in combination with constraints on the matter density parameter. The exact nature, however, of this dark energy is not well understood at present. Vacuum energy or a cosmological constant is the simplest explanation, but attractive alternatives like a dynamical scalar field (quintessence) have also been explored in the literature. An important task nowadays in cosmology is thus to find new methods that could directly quantify the amount of dark energy present in the Universe as well as determine its equation of state and time dependence. New methods may constrain different regions of the parameter space and are usually subject to different systematic errors, and they are therefore crucial to cross-check (or complement) the SNe results. The test we focus on is the one suggested by Alcock and Paczynski (AP) , which has attracted a lot of attention during the last years. We show, through Monte Carlo simulations, that the AP test, as applied to quasar clustering, is a powerful tool to probe the cosmological density and equation of state parameters, W_m0, W_x0 and w. By taking into account the effect of peculiar velocities upon the correlation function we obtain, for the Two-Degree Field QSO Redshift Survey, the predicted confidence contours for the cosmological constant (w = -1) and spatially flat (W_m0 + W_x0 = 1) cases. It turns out that, for w = -1, the test is especially sensitive to the difference  W_m0 - W_L0, thus being ideal to combine with CMB results. We also find out that, for the flat case, it is competitive with future supernova and galaxy number count tests, besides being complementary to them.