M = 0; phi = Pi/2;
t = 1; t2 = 0.5; t3 = 0.35;

b1 = {{-Sqrt[3]/2}, {3/2}};
b2 = {{-Sqrt[3]/2}, {-3/2}};
b3 = {{Sqrt[3]}, {0}};

hx[kx_, ky_] :=
  t ((1 + Cos[{kx, ky}.b1] + Cos[{kx, ky}.b2])) +
   t3 (2 Cos[{kx, ky}.(b1 + b2)] + Cos[{kx, ky}.(b1 - b2)]);
hy[kx_, ky_] :=
  t ((Sin[{kx, ky}.b1] - Sin[{kx, ky}.b2])) +
   t3 Sin[{kx, ky}.(b1 - b2)];
hz[kx_, ky_] :=
 M - 2  t2 Sin[
    phi] (Sin[{kx, ky}.b1] + Sin[{kx, ky}.b2] + Sin[{kx, ky}.b3])
H[kx_, ky_] =
  Flatten[{hx[kx, ky], hy[kx, ky], hz[kx, ky]}, 1] // Simplify;

eqs = Thread[{x, y,
    z} == {t ((1 + Cos[{kx, ky}.b1] + Cos[{kx, ky}.b2])) +
     t3 (2 Cos[{kx, ky}.(b1 + b2)] + Cos[{kx, ky}.(b1 - b2)]),
    t ((Sin[{kx, ky}.b1] - Sin[{kx, ky}.b2])) +
     t3 Sin[{kx, ky}.(b1 - b2)],
    M - 2  t2 Sin[
       phi] (Sin[{kx, ky}.b1] + Sin[{kx, ky}.b2] + Sin[{kx, ky}.b3])}]
impl = GroebnerBasis[Join[TrigExpand[eqs],
     {Cos[(Sqrt[3] kx)/2 - 3 ky/2]^2 +
        Sin[(Sqrt[3] kx)/2 - 3 ky/2]^2 == 1,
      Cos[(Sqrt[3] kx)/2 + 3 ky/2]^2 +
        Sin[(Sqrt[3] kx)/2 + 3 ky/2]^2 == 1,
      Cos[(Sqrt[3] kx)]^2 + Sin[(Sqrt[3] kx)]^2 == 1,
      Cos[(3 ky)]^2 + Sin[(3 ky)]^2 == 1}],
    {x, y, z},
    {Cos[(Sqrt[3] kx)/2 - 3 ky/2],
     Sin[(Sqrt[3] kx)/2 - 3 ky/2],
     Cos[(Sqrt[3] kx)/2 + 3 ky/2],
     Sin[(Sqrt[3] kx)/2 + 3 ky/2],
     Cos[(Sqrt[3] kx)],
     Sin[(Sqrt[3] kx)],
     Cos[(3 ky)],
     Sin[(3 ky)]}][[1]] // FullSimplify