We investigate structural relaxations in La-Al-Ni/Cu metallic glasses and find that the La-Al-Ni glasses tend to have a significant β-relaxation, which becomes more pronounced and shifts to high temperatures when increasing Ni content from 10 to 30 at.% in La85-xAl15Nix or Al content from 10 to 25 at.% in La70AlxNi30-x glasses. In contrast, the β-relaxation in La-Al-Cu glasses is weak and almost disappears after annealing. Compared with La-Al-Cu glasses, La-Al-Ni glasses usually have higher Tg temperatures and bigger changes in Tg by sub-Tg annealing. Different relaxation behaviors between La-Al-Ni and La-Al-Cu glasses changing with compositions and thermal history indicate that the free volume annihilation is not the key factor in controlling the β-relaxation in La-Al-Ni glasses.