The Hook Horrors entry says that they understand the Hook Horror language.
Therefore, you are able to communicate with them using the tongues spell, which specifies that:
...any creature that knows at least one language and can hear the target understands what it says.
Furthermore, any other effects that permit a creature to communicate with any creature that has language will also work (i.e. the monk's Tongue of the Sun and Moon feature, the Great Old One warlock's Awakened Mind, etc.).
This will let you handle communication to the Hook Horror. I'm not sure it will cover the responses as the stipulation is that you understand "spoken languages," and so it might be a DM's call as to whether banging on your chest counts as spoken. All voices are just vibrating vocal chords, so there's some merit to the notion that banging on an exoskeleton and creating deliberate vibrations qualifies as a spoken language.
If the DM rules against that, you would need some sort of two-way telepathy, such as Rary's telepathic bond. You might want to start with tongues, though, so that the creature lets you try the latter.
If you want to get really high level, you can true polymorph into an angel and just know all languages and have telepathy.