Elemental Adept applies to Dragon's Breath when cast on yourself
The Elemental Adept feature doesn't qualify that any spell damage is required to be rolled on the same turn as when the spell is cast. So because the damage from Dragon's Breath comes from a spell, it would qualify for this feature.
The caveat is that this is only absolutely true if the caster targets themselves with the effect of the original spell. If they target someone else, then the ruling is different.
It is ambiguous as to whether other creatures that have Dragon's Breath cast on them could benefit from Elemental Adept
There's two scenarios to consider:
Caster A casts Dragon's Breath on Creature B, Creature B has the Elemental Adept feature
In this scenario, creature B does not benefit from Elemental Adept: the feat specifically requires that the damage come from a spell that the creature cast themselves. Because the effect was conferred upon them by someone else, it would not qualify in this scenario.
Caster A casts Dragon's Breath on Creature B, Caster A has the Elemental Adept feature
This is ambiguous, because when a creature buffed with Dragon's Breath rolls damage dice for their action, it's not clear whether they count as rolling the damage dice or whether it's the original spellcaster who does.
- If, as DM, we adjudicate that Creature B is rolling the damage dice, then these damage rolls would not benefit from Elemental Adept.
- If, however, we adjudicate that Caster A is rolling the damage dice, then these damage rolls would benefit from Elemental Adept.
So while I don't know the correct answer to this particular conundrum (maybe the topic of a different question?), I do believe that if you or your DM decides on the correct answer, then one of the two above statements is correct.