The question refers to a moment in time which lies in the past, in relation to the moment that the question was asked:
At which moment did he start being able to move so quickly?
would be equivalent in meaning to the orignal question.
At which moment in time does he start being able to...
clearly doesn't make sense when referring to a moment in the past. That's the reason why could is semantically more correct then can, even though it's not as obvious when using this form.
On a sidenote: the adjective quick should be replaced by the adverb quickly.
Also note the comment by @p-e-dant, which mentions the use of the modal verb could to highlight a present ability/possibility. This is another possible interpretation, although I find it doesn't completely apply.