Understanding and controlling heat transport in molecular junctions would provide new routes to design nanoscale coupled electronic and phononic devices. Using first-principles full quantum calculations, we tune thermal conductance of a molecular junction by mechanically compressing and extending a short alkane chain connected to graphene leads. We find that the thermal conductance of the compressed junction drops by half in comparison to the extended junction, making it possible to turn on and off the heat current. The low conductance of the off state does not vary by further approaching the leads and stems from the suppression of the transmission of the in-plane transverse and longitudinal channels. Furthermore, we show that misalignment of the leads does not reduce the conductance ratio. These results also contribute to the general understanding of thermal transport in molecular junctions.