Rates of nucleation of two-flavor quark matter in a neutron star core, originally composed of nuclear matter in beta equilibrium, are calculated by using a quantum tunneling analysis incorporating the electrostatic energy and the effects of energy dissipation. We find that low-energy excitations in nuclear matter act to increase the overpressure required to form a quark matter droplet via their collisions with the droplet surface, and that the nucleated droplet would develop into bulk matter due to the screening effects of electrons and muons on the droplet charge. We also address the question of how the presence of hyperons in the hadronic phase affects the deconfinement transition. It is found that strangeness contained in hyperons acts to reduce the effective mass and electric charge of the droplet.