Here is the full list of all 38 It's Quiz Time achievements.
Someone joins mid-game
The same player is leading at the end of every round and wins the game
A new player wins the game
The leader beats a team of two or more players in Showdown
Someone calls 'Shenanigans!' during Describe It!
The same player is chosen as the best friend by all players in a round of Best Friends Forever
Everyone answers correctly in Best Friends Forever, apart from the best friend
Best friend fails to correctly predict player's answer in Best Friends Forever
A player goes from last place after the first round to winning the game
A team of two or more players beats the leader in Showdown
The same player answers every question correctly during All in Order
A player scores over 9,999 points in a multiplayer game
Complete a game with friends
A player correctly answers all specialist subject questions during The Big Finish
A player scores over 450 points on a single question in The Big Finish
All players choose the same specialist subject during The Big Finish
A player answers correctly during Trust or Bust when all other players predicted otherwise
Complete a Solo Score Attack game
Complete 10 games with friends
Complete 30 games with friends
Play two back-to-back games with the same players
Complete a game with 8 players
The same player wins three games in a row
All players answer every question correctly during the Starter Round
No-one answers incorrectly during a round of Connections
The same player answers every question incorrectly during All in Order
A player guesses only one clue during Describe It!
A player correctly guesses every clue for a subject in Describe It!
A clue is removed during Describe It! after someone calls 'Shenanigans!'
No-one answers correctly during Trust or Bust
The same player correctly predicts that everyone will answer correctly during Trust or Bust
The same player predicts everyone will answer incorrectly during Trust or Bust
The same player works alone against a team of two or more players in Showdown 5 times
All players in a 4-player multiplayer game finish with over 7,500 points
Finish a Solo Score Attack game with zero points
Score over 8,000 points in the Solo Score Attack
Play on October 31st
Play on December 25th