Hickey is Brentford's first summer signing.
Brentford have paid an initial £14m for Hickey, which is a club record fee.
The 20-year-old made 48 appearances for Bologna after joining from Hearts in the summer of 2020.
36 of those appearances came in Serie A last season, during which Hickey also added five goals and one assist.
Responding to the completion of the deal, head coach Thomas Frank told the club's official website: "I am very pleased that we have managed to sign Aaron. What he has already achieved in his career is very impressive. He is only 20 and has played a full season in Scotland as well as almost two seasons in Serie A.
"We think that Aaron is a player with great potential, who already has an impressive number of games at a good level under his belt. We believe that we have proven we are a good club for players to maximise their potential. We think we can help him be even better.
"Aaron is a rare thing in modern football in that he is equally good with both feet. He is technically very sound and is good when he has the ball in tight areas. He is an attacking full back that has good creativity, he is a great crosser of the ball, and scored goals in Serie A last year.
"Having played two years in Italy, we know he will have good defensive traits. There are areas to improve and we are looking forward to having him with us so we can be part of his development as a player."
Hickey has won four caps for the Scotland national team.
