Protein-lysine methyltransferase that selectively trimethylates residues in heat shock protein 70 (HSP70) family members. L-lysyl-[protein] + 3 S-adenosyl-L-methionine = 3 H(+) + N(6),N(6),N(6)-trimethyl-L-lysyl-[protein] + 3 S-adenosyl-L-homocysteine Belongs to the methyltransferase superfamily. METTL21 family.