Mesoscale eddies have strong signatures in sea surface height (SSH) anomalies that are measured globally through satellite altimetry. However, monitoring the transport of heat associated with these eddies and its impact on the global ocean circulation remains difficult as it requires simultaneous observations of upper-ocean velocity fields and interior temperature and density properties. Here we demonstrate that for quasigeostrophic baroclinic turbulence the eddy patterns in SSH snapshots alone contain sufficient information to estimate the eddy heat fluxes. We use simulations of baroclinic turbulence for the supervised learning of a deep Convolutional Neural Network (CNN) to predict up to 64% of eddy heat flux variance. CNNs also significantly outperform other conventional data-driven techniques. Our results suggest that deep CNNs could provide an effective pathway towards an operational monitoring of eddy heat fluxes using satellite altimetry and other remote sensing products.