Try this modification.
Notice I changed two {} to ()
gamma = 1; alphag = 0.1; Beff = {0, 0, 10}; z = {0, 0, 1}; J = -1;B = {0, 1, 0};
Heff[t_] := J*{m1x[t],m1y[t],m1z[t]} + B;
Heff1[t_] := J*{mx[t],my[t],mz[t]} + B;
cons[t_] := Cross[{mx[t],my[t],mz[t]}, Heff[t]];
cons1[t_] := Cross[{m1x[t],m1y[t],m1z[t]}, Heff1[t]];
tGilbdamp[t_] := alphag*Cross[{mx[t],my[t],mz[t]}, cons[t]];
tGilbdamp1[t_] := alphag*Cross[{m1x[t],m1y[t],m1z[t]}, cons1[t]];
tmax = 100;
LLGS = {{mx'[t],my'[t],mz'[t]} == -gamma*(cons[t] + tGilbdamp[t]),
{mx[0],my[0],mz[0]} == Normalize[{-Sqrt[3.]/2, -1/2, 0}]};
LLGS1 = {{m1x'[t],m1y'[t],m1z'[t]} == -gamma*(cons1[t] + tGilbdamp1[t]),
{m1x[0],m1y[0],m1z[0]} == Normalize[{Sqrt[3.]/2, -1/2, 0}]};
sol1 = NDSolve[{LLGS, LLGS1}, {mx[t],my[t],mz[t],m1x[t],m1y[t],m1z[t]}, {t,0,tmax}]
That doesn't give any errors and seems to come up with solutions.
Please check that very carefully to make certain that I have not made any mistakes or misunderstood what you were trying to do.