Not really.
The carbon-rich planets described in the Wikipedia article have formed by condensation from gases in which carbon was more abundant than oxygen, which happens in the gases from which some stellar systems have formed.
The Solar System has formed by condensation from gases where there was more oxygen than carbon, and this remains true for Mercury.
Mercury has a somewhat higher proportion of carbon than other planets, because much oxygen has been lost in space, as a component of water and of other easy to vaporize oxides, due to the high temperature of its surface. Even so, it contains much more oxygen in silicates, than carbon.
The core of Mercury is made of iron alloy, like all planet cores. This study supposes that the liquid core has been saturated in carbon. In this case, when the core has solidified, it must have expelled a part of the dissolved carbon at the surface of the core. Due to the high pressure, the expelled carbon must have formed diamond, instead of forming graphite, as it happens at the solidification of smelted pig iron (after a sufficient time, so that any metastable carbides will decompose).