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The aliens in question are highly intelligent (at least smarter than you dumb apes on this rock) and have a language of articulated sounds (specifically, the syllables "Khar" and "Kie") where the pitch and tone show the meaning.
They can control these basically as if they had a human mouth...and yet they have no mouth at all. The only openings on their face-like structure (where the sound does come from) are two "eyes." What alternate ways of articulating could they use to speak with no mouth?
They could make noises within their bodies, which comes out of nowhere in particular. Kind of like when your stomach grumbles, and other people hear it. Except they might have a special organ that makes these noises, like humans have voice boxes. They could accentuate their language as well, perhaps by developing some gestures or tapping rhythm that holds special meaning. And if they are bioluminescent, then that can come into play as well, perhaps by describing how "bright", or good, their mood is.
Insects produce sounds by rubbing their body parts together so they could make sounds by rubbing different parts of their body together or have an organ that is specifically for making sound by having parts that rub together. I'm not sure if this could produce the specific sounds you want though. The sounds this alien would make would sound more like a musical instrument that produces sound from rubbing, scraping, or banging two objects together than to a human talking. This means that while it could produce a rich variety of sounds it may not be able to produce the same exact sounds we can make but could instead make sounds that are similar to the ones we make.
Crickets and many other insects create noise by rubbing body parts together, for example crickets with their wings. Doing this they can produce a wide variety of sounds.
It's unlikely that an alien using a similar method would be able to reproduce the sounds of human speech, but they could certainly produce a wide enough range of different sounds for a fully functional language.
Either they do this by rubbing limbs together but if you want the sounds coming specifically from the heads then they could have the sound producing limbs hidden inside air spaces within their head. Perhaps to protect the delicate sound-producing limbs from damage by shielding them from the world.
Whales recirculate air through a vocal organ and don't use their mouth to make noise.
A great SF example is aliens that use a timpanum for speaking.
When an alien locks eyes with another alien, they can freely communicate with each other until the bond is broken. You can manage that using a physical link (made of light?) generated from glands directly underneath both aliens' eyes. This also makes it easy to have aliens overhear conversations/partake in group conversations. They can make eye contact with the link, and they link with it too, being able to again hear and speak in the conversation.
Another similar idea: You can manage the bond using an invisible link that both aliens sense somehow. This way, nobodyalien can listen in on conversations they shouldn't.
A third idea, and my personal favorite: Combine the two ideas!
The alien initiating the conversation is able to consciously choose what type of connection they make with the person they are talking with. Both ways have their advantages and disadvantages, but the second one is usually more desirable, as privacy tends to be better. Maybe the second option costs more energy/brainpower/whatever to initiate?
TL;DR: Aliens communicate telepathically. 👽
There's a pair of excellent YG Acoustics Anat Reference II speakers I would recommend. At $107,000, they should be a bargain for said ultra-intelligent alien race.
Their precise transducer replicates signals sent from the amplifier with absolute fidelity, ensuring that even dumb Earth-bound apes can hear (enviously) what the aliens wish to communicate in a premium, non-distorted package.
