No, charm person is not obvious
In Chapter 10: Spellcasting of the rules, the Targets section states that spells with subtle effects can be cast without being noticed (PHB, p. 204; emphasis added):
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 the spell says otherwise.
Even the target typically doesn't know
The casting of charm person is subtle enough that even the target doesn't know about the spell until it expires (PHB, p. 221):
When the spell ends, the creature knows it was charmed by you.
If even the target of the spell is unlikely to perceive that it was cast, other observers would have even less of a chance. (A court magician, trained in Arcana and watching for such things, would be a different case.)
Obvious effects are called out in the spell description
Spells that are obvious when cast call that out in the description. For example, knock makes a loud noise that alerts nearby creatures.
Social Spells are described in subtle terms
Compare the description of burning hands:
As you hold your hands with thumbs touching and fingers spring, a thin
sheet of flames shoots forth
...to that of dominate person
You attempt to beguile a humanoid [...]
These descriptions describe what casual onlookers will notice. You can try to charm or beguile someone without magic, it just might not be as effective.
Similar spells provide more detail. The material component of friends is described as:
A small amount of makeup applied to the face as the spell is cast
And after the friends spell expires, the target "realizes you used magic to influence its mood" - calling out the effect of the spell, not the components of the casting, as having become obvious.
Lastly, let's consider suggestion:
You suggest a course of action (limited to a sentence or two) and magically influence a creature
In this case the RAW are pretty clear; the physical manifestation of the spell is just that - there's no imp that appears on the target's shoulder to whisper in its ear.
Thus, when the wizard suggests to the head guard, "These aren't the constructs you are looking for," would we really expect the other guards to recognize the "combination of sounds, with specific pitch and resonance" in the old man's voice, and open fire?
School of Enchantment wizards gain the ability to make targets of their charms unaware of the spell after it was cast, but there's no mention of the spell-casting itself going from obvious to subtle, which one might assume would be part of the enchanter benefits if it were needed.
Also, if casting spells were tremendously obvious, many illusion spells would be not so useful.
Sorcerer's Subtle Spell Metamagic
Sorcerers have a Metamagic option called Subtle Spell, which allows them to spend sorcery points to cast the spell without verbal or somatic components.
This has the power to make otherwise obvious spells subtle. For example, a manacled and gagged spell caster suddenly transforms into a wolf-man and bursts his bonds. The guards think he's a werewolf, but he just used Subtle Spell and cast the alter self and knock spells.