Drum island isn't the homeland
of the cherry blossoms. Hiluluk wanted to bring the cherries to Drum so that the winter island could hope again.
Which means that he first saw the cherries
elsewhere.
So he did leave Drum at least once.
There are also fan theories that the place WB and Roger are shown having sake together during Marineford is the same as Hiluluk's cherries, which in itself could mean that Hiluluk could've traveled with one of them, but that's beyond the point here.
Not only that, but he was a quack doctor who received a different medical education than Drum's own doctors. Meaning, again, that he was medically trained
away from Drum.
As for Doc Q being Hiluluk's son:
- Hiluluk is a
Quack
Doctor. That is his title. It's quite ironic that Doc Q has those initials inverted
DQ.
- Doc Q itself, as a name, is a pun on the Japanese "doku", meaning poison. Doc Q is a poisonous doctor, killing his patients; Hiluluk was given a poisonous mushroom by Chopper by mistake, and died because of it
- Hiluluk's own philosophy is that "A person only dies when their name is forgotten." Doc Q's nickname is Shinigami, Doctor of Death, even carrying a scythe with him
- Hiluluk is paired with Chopper, a reindeer; Doc Q is always with his horse, Stronger
- both Hiluluk and Doc Q have pale faces, big noses, long hair
- both Hiluluk and Doc Q use explosives
- both Hiluluk and Doc Q are sick doctors. Doc Q has been sick and frail since he was a child
- both Hiluluk and Doc Q are criminals: Hiluluk was a thief before becoming a doctor, and Doc Q is a pirate
- Doc Q was both 28 years ago, when Hiluluk was studying the cherry blossoms,
away from Drum
- and Doc Q is among the BBP who attacked Drum specifically
So, on the one hand you have
a quack doctor, former thief, chronically-ill chemist, obsessed with curing disease, and bringing the miracle of the cherry blossoms back to Drum.
On the other hand:
a sickly doctor, pirate, chemist, poison/death-themed doctor who weaponizes disease and who was personally involved in the attack on Hiriluk's homeland.
The parallels couldn't be any clearer.