For HZSM-5, the zeolite ion exchange sites are occupied by H+ ions. This is normally done by ion exchanging the zeolite with NH4Cl to produce a NH4+ exchanged zeolite. That zeolite is then heated to decompose the NH4+ to H+ and ammonia.
ZSM-5 is more general in that any ions can occupy the ion exchange sites. For example, both HZSM-5 and NaZSM-5 are ZSM-5, but it doesn’t work the other way around.
Another term that you will see in the literature for ZSM-5 is MFI.