In 2009, Garver was named the Albuquerque Public Schools Male Athlete of the Year and an all-state and honorable mention All-American in baseball.

[5] As a junior in 2012, he was named co-Mountain West Conference player of the year, sharing the honor with teammate D.J.

[1] The Minnesota Twins selected Garver in the ninth round of the 2013 Major League Baseball draft.

[14] Garver returned to Rochester to start 2017, where he displayed the best power of his pro career thus far, batting .291/.387/.541 with 17 home runs in 88 games.

[12] The Twins promoted Garver to the major leagues on August 18, 2017,[15] and he made his MLB debut the next day, striking out as a pinch hitter.

Garver left the game[21] and was put on the 10-day injured list, replaced on the roster by Miguel Sano.

He was off the Twins roster several times, due to a groin contusion, lower back tightness, and the birth of his first child.

[15][25] On July 27, Garver became a part of major league history when both he and Detroit Tigers catcher Eric Haase hit grand slams.

[28] On March 12, 2022, the Twins traded Garver to the Texas Rangers for Isiah Kiner-Falefa and Ronny Henriquez.

[12] Garver had surgery to repair a torn flexor tendon in his right forearm in July 2022, ending his season.

Though a left knee sprain sidelined him from April 10 to June 2, he hit .270/.370/.500 with 19 home runs in the regular season.

[15] His slugging continued early in the playoffs, as he hit a grand slam in the Rangers 11–8 win over the Baltimore Orioles in Game 2 of the American League Division Series.

Garver homered for the Rangers' only run in a Game 2 loss to the Arizona Diamondbacks but otherwise managed one single and one walk in 22 plate appearances in the Fall Classic.

[33][34] Garver was tabbed as the team's primary DH before the 2024 season,[35] though he caught 24 games backing up Cal Raleigh.

[47] The Garvers created an endowed scholarship at Oregon State in 2021 honoring their late dog, a boxer named Benny who died of lymphoma.