How could lithium burning take place in a quasi-star?
Lithium, along with Hydrogen and Helium, was one of the 3 elements created in the Big Bang. Thus, it should exist to some part in any star that hasn't burnt all of it out, and as mentioned, it's not an easy thing to do. 

Population III stars are expected to contain Lithium, and Beryllium as well. The amount, however, is not particularly high.