Ability Score damage doesn't trigger a Concentration check. Nonlethal damage does.
Looking carefully at the relevant rules, it appears that ability damage doesn't actually damage the character, but rather damages their scores.
Diseases, poisons, spells, and other abilities can all deal damage directly to your ability scores.
Since this damage is affecting a character's scores, and not the character themselves, the damage doesn't count under the Injury section of the Concentration rules.
If you take damage while trying to cast a spell, you must make a concentration check
Nonlethal damage, however, is still damage to a character.
When you take nonlethal damage, keep a running total of how much you’ve accumulated.
Since Injury applies whenever a character takes damage, taking nonlethal damage still requires a Concentration check.