No, not by RAW. However, in a non-AL game, given enough time, you can gain proficiency with a language.
Xanathar's Guide To Everything has rules for this in the Downtime section, under Training. It states the following:
Given enough free time and the services of an instructor, a character can learn a language or pick up proficiency with a tool.
Resources. Receiving training in a language or tool typically takes at least ten workweeks, but this time is reduced by a number of workweeks equal to the character’s Intelligence modifier (an Intelligence penalty doesn’t increase the time needed). Training costs 25 GP per workweek.
You can easily modify this and allow listening in to conversations to count as a few workdays for the purpose of this "downtime" activity. That is, however, not legal in AL (since it'd be a house rule, which are not permitted). Otherwise, your DM still has to explicitly approve this modification (obviously).
Similarly, your DM could rule that (regardless of whether listening to conversations reduces the required downtime) the closer you get to the end of your language-learning-process, the more pieces of conversations you understand. Again, this is not supported by RAW, so it's not legal in AL.
On a side note, bear in mind that, according to XGtE, you should avoid too many breaks in between spending your downtime:
The days of an activity don’t need to be consecutive; you can spread them over a longer period of time than is required for the activity. But that period of time should be no more than twice as long as the required time; otherwise you should introduce extra complications (see below) and possibly double the activity’s costs to represent the inefficiency of the character’s progress.