The Hawks also get a 2015 second-round draft pick in the deal, which was announced on Monday.
The move clears more than $15 million in salary cap space for Atlanta. Salmons has a $1 million buyout of his $7 million salary, creating even more financial flexibility for the Hawks.
Nogueira, the 16th overall pick last year, spent last season in Spain. Williams, who is under contract for almost $5.5 million next season, averaged 10.4 points in 60 games for Atlanta last season.
Nogueira, a defensive-minded 7-footer who had some knee issues last season, has one year left on his Spanish contract.