Does dredge trigger when you would draw during an opponent's turn effect that makes you draw?

Dredge is a replacement effect, not a triggered ability. Whenever you would draw a card, you may replace that draw with the dredge effect — milling cards equal to the dredge number and returning the dredge card to your hand. This applies to any draw you would make, no matter the source or whose turn it is.

CR 702.52a defines dredge: if you would draw a card, you may instead mill N cards and return the dredge card from your graveyard to your hand. CR 614.1 establishes that replacement effects modify events as they would occur, and CR 614.2 clarifies they apply any time the event would happen — there is no restriction on turn or source.

So if an opponent plays a card like Howling Mine or Windfall that causes you to draw during their turn, you may still choose to replace each of those draws with a dredge activation (provided you have a legal dredge card in your graveyard and enough cards in your library).

Example: It's your opponent's turn and they activate Staff of Domination to make you draw a card. You have Golgari Grave-Troll (Dredge 6) in your graveyard and at least 6 cards in your library. Instead of drawing, you may mill 6 and return the Grave-Troll to your hand.

Note: If your library has fewer cards than the dredge number, you cannot use dredge for that draw (CR 702.52a) — the draw happens normally (or you lose if your library is empty, per CR 104.3c).