It's not under any influence of alcohol! It's just a gesture. Eyes are half closed with creases on the forehead and some tension in the muscles around the eyes. Such expressions are made when you doubt something or find something suspicious.
I'm trying a lot but not finding even a single image! Why? Because I don't know what it is called!
In detective stories, you'll often read the expression
he narrowed his eyes
If your eyes narrow or if you narrow your eyes, you almost close them, for example because you are angry or because you are trying to concentrate on something.
More often than not, when someone is thinking deeply, English speakers will focus on the creases in the forehead, which is called a frown
When someone frowns, their eyebrows become drawn together, because they are annoyed, worried, or puzzled, or because they are concentrating.
Perhaps the word you are looking for is to squint but it does not imply that you doubt something, you squint because you want to look at something with your eyes partly closed in order to see better (LDOCE 5th version).
But here's an alternative for squint:
to narrow one's eyes - to partly close them, especially to show that you do not trust someone
One oft-used verb for this is squint:
squint (v.) To look with the eyes partly closed, as in bright sunlight, or as a threatening gesture; to look askance, as in disapproval.
The word is often used in conjunction with suspicion. A Google Books search yielded a heap of entries with squint suspiciously, such as:
The village, Burns thought, had the look of a distrustful old man squinting suspiciously at all outsiders.
People from the parking lot at the front of the store began wandering over, staring or squinting suspiciously.
When he left the head table, he cast a squinting, suspicious gaze around the arena.
His eyes were small wet holes pressed between a couple dozen layers of wrinkles, probably from a lifetime of squinting suspiciously at people.
I believe you mean that look that Fry had when he was in doubt his friend Bender was impersonated by someone else:
This is indeed squinting, and it is a recognisable expression of suspicion in some cultures, but not universally. In general, squinting means just that:
to look at someone or something with one or both eyes partly closed in an attempt to see more clearly or as a reaction to strong light.
There are some good answers here already, but another couple of terms that come to mind are furrowing one's brow or knitting one's [eye]brow[s]. Furrowing describes an expression where you create wrinkles on your forehead/brow (as one creates furrows in the ground when plowing it) either by raising your eyebrows or tensing the muscles around the eyes in the way you describe. Knitting one's brows refers more specifically to pulling your eyebrows together, which also produces the effect you described.
Edit: I just found an interesting list of terms/expressions describing various facial expressions: https://www.macmillandictionary.com/us/thesaurus-category/american/to-make-a-particular-facial-expression
Similar facial expression for a scowl, but scowl has slightly more angry connotation.
scowl (v.) to look at someone or something in a way that shows anger or disapproval
scowl (n.) an expression on someone's face that shows anger or disapproval
Not mentioned so far: to glower.
glower (v): Have an angry or sullen look on one's face; scowl.
Glower is appropriate if you narrow your eyes in anger or threat. As an illustration of the difference between looking sleepy, squinting, and glowering, see this clip from the movie Get Shorty
There are some kinds of half-shut eyes, here are some of them:
1) While a person is very tired and he is about to sleep, or alternatively before death or if having a structure of such eyes. It can be called simply "half-shut eyes" or "half closed eyes"
I saw his half-shut eyes. (reference)
N.B. Sometimes it may be called: piggy-eyed or "pink eye".
2) While suspecting in something or having a concern or a thought about something: half-shut eyelids or "half closed eyes".
"A woman at the information desk with half-shut eyelids managed to tell me that the KLM flight was delayed with no ETA." (reference)
The officer looked at him narrowly through half-closed eyes... (reference)
3) While a person want to improve his eyesight (by changing the shape of our eye and letting in a limited amount of light that is more easily focused) : squint or less common blink (see definition No.1)
Why can people see more clearly when they squint their eyes? (reference)
4) While conveying a glance: "bedroom eyes"
Marilyn Monroe's signature look was her bedroom eyes. (reference)
That half-closed 'bedroom eyes' look actually makes women think you are SHIFTY, not sexy. (reference)
5) Pathological disability to close the eyes completely (for example due to a malfunction in a facial nerve): Lagophthalmos (or Nocturnal lagophthalmos if it is at night)
He has Lagophthalmos. He can't close his eyes completely.
So in the end of the day, the choice depends on context. But in your case I would consider the option No.2 here.
Narrow your eyes: to squeeze the eyelids closer together.
Squint: look at someone or something with one or both eyes partly closed in an attempt to see more clearly or as a reaction to strong light.
Cock your eye:glance in a quizzical or knowing manner with a raised eyebrow.
Squinch:to press together the features of the face.
Be askance:to look at or think about someone or something with doubt, disapproval, or no trust.
Screw your eyes:to tighten the muscles of (the face or eyes).
