Does the Bestow Curse spell’s extra damage apply to the Dragon’s Breath spell?

The bestow curse spell states:

[…] While the target is cursed, your attacks and spells deal an extra 1d8 necrotic damage to the target. […]

I’m wondering how this interacts with the dragon’s breath spell which grants a creature you touch a new action to create a damage-dealing area of effect. Does this count as your spell dealing damage and thus activate hex? If note, does this change if dragon’s breath was originally cast on yourself?