For starters, you're going to want to have 3 friends to play this game with or make a session. Finding 3 randoms all in 1 game and then doing 5 quests with them is extremely rare.
Some things I'd like to mention first. It's not 5 consecutive quests meaning you can do 2 quests one day with 3 other people and then 3 quests another day with 3 other people and you will get the achievement.
If someone disconnects from the game, wait for them to join back before completing a quest. As long as you have 3 other people in your game (4 is the max you can have in a game), you will get progress towards your 5 quests completed when turning in a quest.
If you want to make it easier, you can do 5 quests without actually turning them in by yourself. Then once you get 3 other players in your game, turn the 5 quests in. Make sure they're in the game and not joining.
Connecting may be a problem. If it becomes an issue, everyone reset their Xbox and see if this works.