The accretion behaviour in low-mass X-ray binaries (LMXBs) at low luminosities, especially at <1034 erg s-1, is not well known. This is an important regime to study to obtain a complete understanding of the accretion process in LMXBs, and to determine if systems that host neutron stars with accretion-heated crusts can be used probe the physics of dense matter (which requires their quiescent thermal emission to be uncontaminated by residual accretion). Here, we examine ultraviolet (UV) and X-ray data obtained when EXO 0748-676, a crust-cooling source, was in quiescence. Our Hubble Space Telescope spectroscopy observations do not detect the far-UV continuum emission, but do reveal one strong emission line, C IV. The line is relatively broad (≳3500 km s-1), which could indicate that it results from an outflow such as a pulsar wind. By studying several epochs of X-ray and near-UV data obtained with XMM-Newton, we find no clear indication that the emission in the two wavebands is connected. Moreover, the luminosity ratio of LX/LUV ≳ 100 is much higher than that observed from neutron star LMXBs that exhibit low-level accretion in quiescence. Taken together, this suggests that the UV and X-ray emission of EXO 0748-676 may have different origins, and that thermal emission from crust-cooling of the neutron star, rather than ongoing low-level accretion, may be dominating the observed quiescent X-ray flux evolution of this LMXB.