Carter, who has one season of eligibility remaining, announced his transfer destination Sunday night on Twitter.
He started Texas Tech's season opener but sustained an ankle injury and then assumed a reserve role after he returned. Carter also started the Red Raiders' regular-season finale and remained with the team through spring practice before announcing last month that he would enter the transfer portal.
A junior college transfer, Carter had 677 pass yards with four touchdowns and four interceptions in two seasons at Texas Tech.
Rutgers coach Chris Ash did not name a definitive starting quarterback after spring practice but said sophomore Artur Sitkowski ended the session as the team's No. 1 signal-caller. Sitkowski started 11 games in 2018 and struggled, completing 49 percent of his passes with 18 interceptions and four touchdowns.
The Scarlet Knights are still awaiting a decision on Boston College transfer Johnny Langan, who is seeking immediate eligibility. They also added quarterback Cole Snyder in the 2019 class.