There are no generalised rules for ability score decreases, reductions, penalties, or damage. Every effect that reduces an ability score is highly specific, and sets its own rules and conditions. It might be worth mentioning that (as far as I'm aware), no such effect uses the term "ability score damage".
For examples, the Shadow can reduce Strength, and said reduction lasts until the target finishes a short or long rest.
On the other hand, the Intellect Devourer can reduce Intelligence to 0, and does not specify a way to regain Intelligence, so only effects like Greater Restoration that specifically remove reductions to ability scores would be effective here.
And for something completely different, the Feeblemind spell reduces Intelligence and Charisma to 1, specifying that the target can be fixed by making a saving throw, or any of Greater Restoration, Heal, or Wish.
So the short answer is that it depends entirely on the source of the ability score reduction. However, the only general solutions appear to be Greater Restoration and Wish.
There are a very few things actual scores are used for… but they're nothing that needs scores from a design perspective, since they could instead be based on the modifier, right? The reality is that they exist only as a nod to, or (depending on how you look at it) a holdover from D&D itself.
Why? If I recall the conversation correctly, Adam and Sage felt that Dungeon World wouldn't feel enough like D&D with just the modifiers, so they were kept to help preserve the intended D&D aesthetic.
For someone like your friend who doesn't have prior experience with D&D's mechanical aesthetics it will be mostly lost on them. That's ok, and you can explain it away as a holdover and nod to the game that defined the experience Dungeon World is attempting to distill.
Best Answer
No, ability score damage isn't in 4e
Checking the final version of the online compendium, there's no glossary entry or text reference to ability score damage or drain, which is a pretty good indicator that it's completely absent. I've not manually checked every monster, but this confirms my personal experience of running 4e for about 10 years, and my understanding of this edition's design goals.
For better or worse, ability score damage just does not blend well with the kind of game 4e wants to be, so it was omitted from this edition. It's notably the same for level drain, spell drain, and similar "change your baseline so you have to re-calculate everything on the fly" effects.
(Likewise permanent versions of these effects. 4e is very focused on keeping the party in balance with one another, and one character losing stats or levels forever could easily compromise that.)
4e-Style alternatives
You mentioned healing surge drain as a replacement. Often, classic monsters which traditionally drain levels or ability scores have a similarly-themed debuff effect in 4e. For example, monsters that used to do dex damage might get a slow or immobilise effect, monsters that used to damage strength might impose an attack penalty or a weakened condition.
However, as a rule these won't be permanent—4e just doesn't roll that way. If the potential to deal permanent harm is important to you, a disease delivered during combat might be a closer equivalent.