Can diamonds conduct electricity?

No because they have a tetrahedron structure made by covalent bonds between carbon atoms. The crystal structure has no free electrons that can flow or travel between potential differences. read more

Boron can make diamonds brown or blue and those definitely conduct electricity. Old school and low priced electrical conductivity moissanite testers will identify such diamonds as moissanite (interesting tid bit for diamond buyers that are not gemologists - most don't realize this is an issue). read more

The thermal conductivity of natural diamond is around 22 W/(cm·K), which makes diamond five times better at conducting heat than copper. The high thermal conductivity may be used to distinguish diamond from cubic zirconia and glass. read more