Yes, garlic can increase glutathione levels.
According to studies, allium vegetables, which include garlic, shallots, and onions, can boost glutathione levels, likely due to their sulfur-containing compounds. A number of human and animal studies have found that eating sulfur-rich vegetables may reduce oxidative stress by increasing glutathione levels.
In summary, garlic, being an allium vegetable rich in sulfur, is associated with increasing glutathione levels, potentially reducing oxidative stress.