ESPN's Adrian Wojnarowski was the first to report the deal.
The trade provides Portland with financial flexibility, as Crabbe still has three years remaining on his contract with an annual average salary of over $18.7 million, per Spotrac.
Brooklyn signed Crabbe to an offer sheet last year, which was matched by the Blazers. According to Wojnarowski, the teams had to wait one year after the offer sheet was matched to make a deal involving Crabbe.
Crabbe also reportedly waived his trade kicker as part of the transaction.
The 25-year-old Crabbe was a second-round pick by the Cleveland Cavaliers out of California in 2013, and he was immediately dealt to Portland.
Last season was Crabbe's best in the NBA as he averaged a career-high 10.7 points per game to go along with 2.9 rebounds and 1.2 assists.
He also shot 46.8 percent from the field and 44.4 percent from beyond the arc, both of which were personal bests.
Crabbe will join a rebuilding Nets team for which he could potentially start after almost exclusively coming off the bench in Portland.
The 27-year-old Nicholson was a first-round pick by the Orlando Magic out of St. Bonaventure in 2012.
During his five-year NBA career, he has played for the Magic, Washington Wizards and Nets, averaging 6.0 points and 3.0 rebounds per contest.