This is answered, if not very directly, in the Sage Advice Compendium (version 2.3, if it matters).
Do you always know when you’re under the effect of a spell?
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.
Apparently, what the caster says is not part of "casting the spell".
Generally speaking, if a spell's text defines some activity that is required to use a spell, you still have to do that thing even if you can avoid using components of that type.
For example, if you have to touch the target of a spell, removing the somatic component doesn't remove the need to touch the target. By the same logic, removing the verbal component of suggestion doesn't remove the need to speak the suggestion; it just means you don't have to intone any words of power along the way, or present it in such a way that it is obviously spellcasting.
So judging by the Sage Advice example, you do have to speak, but you can speak conversationally, and there would be no clear indication that you're doing magic to either the target or the bystanders. (They still might guess that you'd used magic if you said something really absurd and the target immediately went along with it, though.)
There's still a material component, so you'd technically need to have the components in hand or be holding an arcane focus, but since there's no somatic component to suggestion, you don't have to go waving it around -- just hold it. Your DM might call for a check to palm the components or ready your arcane focus without being too obvious about it, but that depends entirely on the DM, the situation, and the nature of the tool you're using. Pulling a wand or orb out of your robes might be a lot more difficult to hide than using a staff you're always holding or leaning on, and if the item is already in hand it wouldn't necessarily be apparent that you're now using it rather than just holding it.