You could, if the tailer knows how (it's a kind of specific alteration). If you have a local tailor who makes shirts, they can handle it.
Barrel cuffs can be converted to single French cuffs or convertible cuffs, or the cuffs can just be replaced (usually that involves turning the shirt into a contrast cuff & collar shirt, since an exact color match is impossible, so the tailor just installs white cuffs and collar... depends on the color of your dress shirt).
Depending on circumstances, it's probably a $20-30 job? Worth considering just buying another shirt, one with native French cuffs?