The Subtle Spell Metamagic can be applied through a familiar
This is clear from the fact that you are still the one casting the spell, which is what Metamagic requires:
[…] When you cast a spell, you can spend 1 sorcery point to cast it without any somatic or verbal components. […]
[…] Finally, when you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell. […]
You are still the one casting the spell, as this only triggers when you cast a spell, and the use of “as if” means that it is not actually the case that the familiar is the one casting the spell.
Your familiar may stop being Hidden because the GM says so
Notably, spells cast through your familiar are all going to be touch-range spells, and as such, your familiar must actually touch your target. This would likely end their Hidden status. To quote the “Hiding” section of the rules:
The DM decides when circumstances are appropriate for hiding. […]
[…] In combat, most creatures stay alert for signs of danger all around, so if you come out of hiding and approach a creature, it usually sees you. However, under certain circumstances, the DM might allow you to stay hidden as you approach a creature that is distracted, allowing you to gain advantage on an attack roll before you are seen. […]
Even just approaching a creature so that you can touch it is likely to end up with your revealing yourself, let alone actually targeting it with a touch-range spell.
So while, yes, you can apply the Subtle Spell Metamagic to touch-range spells cast through your familiar from find familiar, it is going to be up to the GM whether your familiar remains Hidden during/after this action.