It seems that at the end of every NFL game the two opposing QBs meet in the field, hug, and chat for a few minutes. My question is, has the league ordered or encouraged them to do this, or is it totally spontaneous each and every time?
Most sports have a tradition of opponents shaking hands or even hugging at the end of a competition, but the meeting between opposing quarterbacks usually seems to get more attention than the other 120+ people meeting on the field at the end of games. The quarterback is usually a captain of the team and the face of the franchise (as well as the highest paid player on the team most often), so we are naturally drawn to their interactions. To directly answer your question, the National Football League has no mandate for players meeting each other on the field after the game. This interaction is simply ingrained in the game and players from a young age.
This SB Nation Article from a few years ago gets into the tradition and how much attention is paid to when this interaction doesn't occur, but confirms it is permissible to skip it.