It may notice the spellcasting; beyond that, maybe not.
An official ruling in the Sage Advice Compendium actually uses suggestion as an example when it discusses whether you know when you're under the effect of a spell:
Do you always know when you’re under the effect of a spell?
You’re aware that a spell is affecting you if it has a perceptible effect or if its text says you’re aware of it (see PH, 204, under “Targets”). Most spells are obvious. For example, fireball burns you, cure wounds heals you, and command forces you to suddenly do something you didn’t intend. Certain spells are more subtle, yet you become aware of the spell at a time specified in the spell’s description. Charm person and detect thoughts are examples of such spells.
Some spells are so subtle that you might not know you were ever under their effects. A prime example of that sort of spell is suggestion. Assuming you failed to notice the spellcaster casting the spell, you might simply remember the caster saying, “The treasure you’re looking for isn’t here. Go look for it in the room at the top of the next tower.” You failed your saving throw, and off you went to the other tower, thinking it was your idea to go there. You and your companions might deduce that you were beguiled if evidence of the spell is found. It’s ultimately up to the DM whether you discover the presence of inconspicuous spells. Discovery usually comes through the use of skills like Arcana, Investigation, Insight, and Perception or through spells like detect magic.
The second paragraph of this official ruling points out that if a creature fails to notice the casting of the spell, it becomes much harder to figure out that suggestion was cast on it; in that case, all it knows is that the caster told it to do something, and it did. Situationally, it might suspect nothing if the suggested course of action was fairly normal for it; the more questionable or unusual the suggested action, the more likely it may be to question or investigate why it would have done such a thing.
If the creature does notice the spell being cast, of course, then it's far more likely to pinpoint the caster as the source of its strange behavior.
The PHB and basic rules include just a few sentences about whether a creature knows it was targeted by a spell:
Unless a spell has a perceptible effect, a creature might not know it was targeted by a spell at all. An effect like crackling lightning is obvious, but a more subtle effect, such as an attempt to read a creature's thoughts, typically goes unnoticed, unless a spell says otherwise.
Xanathar's Guide to Everything provides a clarification on how one can perceive a caster in the process of spellcasting (p. 85):
Many spells create obvious effects: explosions of fire, walls of ice, teleportation, and the like. Other spells, such as charm person, display no visible, audible, or otherwise perceptible sign of their effects, and could easily go unnoticed by someone unaffected by them. As noted in the Player’s Handbook, you normally don’t know that a spell has been cast unless the spell produces a noticeable effect.
But what about the act of casting a spell? Is it possible for someone to perceive that a spell is being cast in their presence? To be perceptible, the casting of a spell must involve a verbal, somatic, or material component. The form of a material component doesn’t matter for the purposes of perception, whether it’s an object specified in the spell’s description, a component pouch, or a spellcasting focus.
If the need for a spell’s components has been removed by a special ability, such as the sorcerer’s Subtle Spell feature or the Innate Spellcasting trait possessed by many creatures, the casting of the spell is imperceptible. If an imperceptible casting produces a perceptible effect, it’s normally impossible to determine who cast the spell in the absence of other evidence.
Suggestion has both verbal and material components. The sorcerer's Metamagic option Subtle Spell can remove verbal and somatic components, but not material components; thus, if the target creature can see you when you cast the spell, it will perceive your spellcasting (though whether it recognizes it as spellcasting might depend on its familiarity with magic/spellcasting).
One way to avoid your spellcasting being perceived is to Ready the spell from behind cover/obscuration, so that the creature does not see you cast it, then step out so you can see the creature and release the spell's energy. You only need to meet the targeting requirements when the spell's energy is released. (This tactic can also be used to prevent counterspelling.) Since the spell's range is only 30 feet, however, you will likely need to use Subtle Spell (or cast in an area of the silence spell or similar) to avoid the verbal component being heard.