There are several occasions that can occur during a game when a character may wish to hold their breath. Unfortunately, not all systems provide rules for this. Of course, you as DM, can create your own rules for these circumstances, but then I wouldn't have anything to write about today.
In our home campaign, we use the following rule. A character can hold their breath for one (1) round (minute), plus one (+1) round per point of Constitution over 10. Thus, a character with a Constitution of 15, could hold their breath for six (1 + 5) rounds (minutes). Characters are not penalized for having lower Constitution scores. Characters with a Constitution of 10 or less can still hold their breaths for at least one round.
A character holding their breath, may do so and still perform non-strenuous tasks, such as walking (or swimming slowly), carrying light loads, picking locks, etc.
Character that are running, carrying heavy loads, fighting, etc., can only hold their breath for half the time (divide by 2 and round up), with a minimum of one round.
Characters that attempt to hold their breaths longer, must roll a Save or they become unconscious that round and will then die the following round. Each consecutive round beyond accumulates a -2 penalty to this Save. Forcing yourself to hold your breath can be fatal. Of course, a healer can attempt to revive a "mostly dead" character, if the DM agrees.
Characters that are doing nothing at all, but holding their breath, may do so for twice as long.
As DM, you may rule that some characters, such as those that are excellent swimmers, clean outhouses for a living, etc., may hold their breaths twice as long because of practice. Yes, this means an excellent swimmer, doing nothing but holding their breath, can hold their breath four times longer overall.