Yes. FindInstance will give the solution:
In[33]:= matrix = {{1/4 (6 x + Sqrt[2] Sqrt[1 - 2 x^2]),
1/4 (-2 x + 3 Sqrt[2] Sqrt[1 - 2 x^2]),
1/2}, {1/4 (6 x - Sqrt[2] Sqrt[1 - 2 x^2]),
1/4 (2 x + 3 Sqrt[2] Sqrt[1 - 2 x^2]), 1/2}, {x,
Sqrt[1 - 2 x^2]/Sqrt[2], 1}};
equation = 0 == Numerator@Together[(1 - 2 (a/b)^2)/2 - (n/m)^2]
sol = FindInstance[equation && a b m n != 0, {a, b, m, n}, Integers]
matrix /. x -> a/b /. sol
Out[34]= 0 == -2 a^2 m^2 + b^2 m^2 - 2 b^2 n^2
Out[35]= {{a -> -1, b -> -2, m -> -2, n -> -1}}
Out[36]= {{{1, 1/2, 1/2}, {1/2, 1, 1/2}, {1/2, 1/2, 1}}}
But Solve still can't solve this problem:
In[31]:= sol = Solve[equation && a b m n != 0, {a, b, m, n}, Integers]
matrix /. x -> a/b /. sol
During evaluation of In[31]:= Solve::svars: Equations may not give solutions for all "solve" variables.
Out[31]= {{n ->
ConditionalExpression[-(Sqrt[-2 a^2 m^2 + b^2 m^2]/(
Sqrt[2] Abs[b])), Or[
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], a >= 1,
b > 2^Rational[1, 2] (a^2)^Rational[1, 2], m <= -1],
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], a >= 1, m >= 1,
b > 2^Rational[1, 2] (a^2)^Rational[1, 2]],
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], a >= 1,
b < -2^Rational[1, 2] (a^2)^Rational[1, 2], m >= 1],
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], a >= 1,
b < -2^Rational[1, 2] (a^2)^Rational[1, 2], m <= -1],
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], m >= 1, a <= -1,
b > 2^Rational[1, 2] (a^2)^Rational[1, 2]],
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], b < -2^Rational[1, 2] (a^2)^Rational[1, 2],
m >= 1, a <= -1],
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], b < -2^Rational[1, 2] (a^2)^Rational[1, 2],
a <= -1, m <= -1],
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], a <= -1,
b > 2^Rational[1, 2] (a^2)^Rational[1, 2], m <= -1]]]}, {n ->
ConditionalExpression[Sqrt[-2 a^2 m^2 + b^2 m^2]/(Sqrt[2] Abs[b]),
Or[
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], a >= 1,
b > 2^Rational[1, 2] (a^2)^Rational[1, 2], m <= -1],
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], a >= 1, m >= 1,
b > 2^Rational[1, 2] (a^2)^Rational[1, 2]],
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], a >= 1,
b < -2^Rational[1, 2] (a^2)^Rational[1, 2], m >= 1],
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], a >= 1,
b < -2^Rational[1, 2] (a^2)^Rational[1, 2], m <= -1],
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], m >= 1, a <= -1,
b > 2^Rational[1, 2] (a^2)^Rational[1, 2]],
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], b < -2^Rational[1, 2] (a^2)^Rational[1, 2],
m >= 1, a <= -1],
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], b < -2^Rational[1, 2] (a^2)^Rational[1, 2],
a <= -1, m <= -1],
And[
Element[
Alternatives[
a, b, m,
2^Rational[-1, 2] ((-2) a^2 m^2 + b^2 m^2)^Rational[1, 2]/Abs[
b]], Integers], a <= -1,
b > 2^Rational[1, 2] (a^2)^Rational[1, 2], m <= -1]]]}}
Out[32]= {{{1/4 (Sqrt[2] Sqrt[1 - (2 a^2)/b^2] + (6 a)/b),
1/4 (3 Sqrt[2] Sqrt[1 - (2 a^2)/b^2] - (2 a)/b), 1/
2}, {1/4 (-Sqrt[2] Sqrt[1 - (2 a^2)/b^2] + (6 a)/b),
1/4 (3 Sqrt[2] Sqrt[1 - (2 a^2)/b^2] + (2 a)/b), 1/2}, {a/b, Sqrt[
1 - (2 a^2)/b^2]/Sqrt[2],
1}}, {{1/4 (Sqrt[2] Sqrt[1 - (2 a^2)/b^2] + (6 a)/b),
1/4 (3 Sqrt[2] Sqrt[1 - (2 a^2)/b^2] - (2 a)/b), 1/
2}, {1/4 (-Sqrt[2] Sqrt[1 - (2 a^2)/b^2] + (6 a)/b),
1/4 (3 Sqrt[2] Sqrt[1 - (2 a^2)/b^2] + (2 a)/b), 1/2}, {a/b, Sqrt[
1 - (2 a^2)/b^2]/Sqrt[2], 1}}}
Furthermore, is there a general approach to dealing with this type of problem rather than just an analysis based on one of its specific elements?