Quiescent prominences host a diverse range of flows, including Rayleigh-Taylor instability driven upflows and impulsive downflows, and so it is no surprise that turbulent motions also exist. As prominences are believed to have a mean horizontal guide field, investigating any turbulence they host could shed light on the nature of magnetohydrodynamic (MHD) turbulence in a wide range of astrophysical systems. In this paper we have investigated the nature of the turbulent prominence motions using structure function analysis on the velocity increments estimated from Hα Dopplergrams constructed with observational data from Hinode Solar Optical Telescope (SOT). The probability density function of the velocity increments shows that as we look at increasingly small spatial separations the distribution displays greater departure from a reference Gaussian distribution, hinting at intermittency in the velocity field. Analysis of the even order structure functions for both the horizontal and vertical separations showed the existence of two distinct regions displaying different exponents of the power law with the break in the power law at approximately 2000 km. We hypothesise this to be a result of internal turbulence excited in the prominence by the dynamic flows of the system found at this spatial scale. We found that the scaling exponents of the $p$ order structure functions for these two regions generally followed the $p$/2 (smaller scales) and $p$/4 (larger scales) laws that are the same as those predicted for weak MHD turbulence and Kraichnan-Iroshnikov turbulence respectively. However, the existence of the $p$/4 scaling at larger scales than the $p$/2 scaling is inconsistent with the increasing nonlinearity expected in MHD turbulence. We also found that as we went to higher order structure functions, the dependence of the scaling exponent on the order $p$ is nonlinear implying that intermittency may be playing an important role in the turbulent cascade. Estimating the heat ing from the turbulent energy dissipation showed that this turbulence would be very inefficient at heating the prominence plasma, but that the mass diffusion through turbulence driven reconnection was of the order of 10$\mathrm{<mrow\; data-mjx-texclass="ORD">10</mrow>}$ cm$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ s$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ . This is of similar order to that of the expected value of the ambipolar diffusion and a few orders of magnitude greater than Ohmic diffusion for a quiescent prominence.