No hydrogen bond will not occur between Cl and H.
A hydrogen bond has an electrostatic component, but what differentiates it from a dipole-dipole interaction is that it also has a covalent component. The HOMO (highest occupied molecular orbital), the nonbonding electrons, of the donor mixes with the LUMO (lowest unoccupied molecular orbital) of the hydrogen atom, which would be the sigma antibonding orbital.
There is no (or very little) orbital mixing between the HOMO of chlorine and the LUMO of hydrogen. This is because chlorine is in row 3. As you move down the rows, the difference in the size between the hydrogen LUMO and the acceptor orbitals becomes larger resulting in poor overlap and less mixing (all the good acceptors are in period 2). Without orbital mixing, there is no hydrogen bond - only a dipole-dipole interaction.
I hope this helps :)