Yes, of course, the reaction 4Li + SiO2 = 2Li2O + Si is not only possible, but also inevitable. Even the smallest amounts of lithium destroy the quartz glass tube because lithium oxide catalyzes the crystallization of the quartz glass into cristobalite.

Just to expand a bit on Vadim Verlotski 's answer: The Li oxide will further react with the excess silica, forming lithium silicates, and that's the melt that causes devitrification of the quartz.