In a Two-Headed Giant game, this ability will potentially trigger twice at the beginning of the opposing team’s upkeep — once for each player on that team.

Hellfire Mongrel’s triggered ability has an “intervening ‘if’ clause.” That means (1) the ability won’t trigger at all unless the opponent whose turn it is has two or fewer cards in hand as their upkeep starts, and (2) the ability will do nothing if that player has three or more cards in hand by the time it resolves.