Yes, Metamagic may be applied to spells that are cast using a class feature instead of spell slots.
The Metamagic feature says a metamagic option may be applied when the sorcerer casts a spell:
You can use only one Metamagic option on a spell when you cast it, unless otherwise noted.
That is the only criteria: the Sorcerer must cast the spell.
Metamagic would not apply if the spell's effect were applied without being cast. For example, some potions (e.g. the potion of clairvoyance, potion of diminution, and potion of gaseous form, to name a few) say things like "gain the effect of" a spell. In such instances, no spell is actually cast, and so Metamagic could not be applied.
Then we refer to the Shadow Magic sorcerer's Eyes of the Dark feature (XGtE, p. 51):
[...] you learn the darkness spell [...]. In addition, you can cast it by spending 2 sorcery points or by expending a spell slot. If you cast it with sorcery points, you can see through the darkness created by the spell.
This tells us that the Sorcerer is casting the spell. This satisfies the criteria as laid out above. As a result, the Shadow Sorcerer may cast darkness for 2 sorcery points and then apply any relevant Metamagic option of their choice for its corresponding additional sorcery point cost to that spell.
In your case, that means a quickened Eyes of the Dark darkness spell would cost 4 sorcery points (and not cost a spell slot), as you say.