Does the polar head group of the phospholipid carry a charge? if so, positive of negative?

A phospholipid is a membrane lipid that has a polar head and two hydrophobic (non-polar) tails. Thus, it is an amphipathic compound.

Yes, the polar head groups of the phospholipid carry a charge and that is why these are soluble in water. These carry negative charge due to the presence of phosphate group and glycerol. The hydrophobic tail is uncharged.