In summary the electricity is coming, not from the lemon, but from a chemical reaction resulting from the differences in electro-negativity between zinc and copper.
Inside the quartz is a crystal lattice of one silicon for every two oxygen atoms, which forms a network of covalent bonds, with electrons being more attracted to the oxygen due to their electronegativity.
And a really strong version of that is between hydrogen and really electronegative atoms, which are ones that like to take the electron density away, like oxygen.
And this is typical when you're dealing with two elements that aren't very different in terms of their electronegativity or their desire to attract electrons.
That actually leads to some extinctions on our planet because oxygen is very good at grabbing onto, it's highly electronegative, and so it turns a reducing kind of atmosphere into an oxidizing atmosphere.
If two atoms in a bond have very different electronegativities, like say hydrogen at 2.1 and oxygen at 3.5, the electrons are more attracted to the atom with the higher electronegativity.