It can try to, against medium sized opponents
To push a character directly up, the Gazer needs to be underneath the character, as it can only push them directly away from itself. So it would need to enter the characters space to move underneath the character. According to p. 191, PHB, Moving Around Other Creatures
You can move through a nonhostile creature’s space. In contrast, you can move through a hostile creature's space only if the creature is at least two sizes larger or smaller than you. [...] Whether a creature is a friend or an enemy, you can’t willingly end your move in its space
A gazer is Tiny. Player characters are Small or Medium. So it cannot enter the space of a Small character, but it can enter the space of a Medium character, because it is at least two sizes smaller than that character. The rules allow the gazer to break up its movement (p. 190 PHB)
You can break up your movement on your turn, using some of your speed before and after your action
The rule does not say you cannot attack or use abilities while moving through another creature's space, just that you cannot end your move there. So, the gazer could move underneath a Medium sized character, and use its ray to push it up 30 feet. Note however that the gazer has no choice of picking the telekinetic ray. It has to roll a d4 to determine the kind of ray it will shoot.
If it manages to throw the character up, the character will then immediately fall down 30 feet again, for 3d6 of damage. There are no rules about how to handle falling onto another creature in the core rules. There is an optional one in Tasha's Cauldron of Everything, Falling onto a Creature (p. 170), but it does not apply if one of the two creatures is Tiny, like the gazer is, so the gazer would not run the risk to take half of that damage.
P.S. The gazer would not necessarily suffer an opportunity attack (p. 195, PHB)
You can make an opportunity attack when a hostile creature that you can see moves out of your reach [...] You also don’t provoke an
opportunity attack when you teleport or when someone or something moves you without using your movement
When the gazer enters the character's space, that does not trigger an opportunity attack. And when you leave the gazer's space (and it yours) while you are being thrown up, the gazer does not use its own movement to move out of your reach, so there's no opportunity attack either. If the gazer can move to a space next to you after pulling off this stunt, it will not leave your reach either, so no opportunity attack there.
P.P.S. Gazers have an Intelligence score of 3 (-4 modifier). I wouldn't exactly call them "smart".