For your original example, [`TrigToExp`](https://reference.wolfram.com/language/ref/TrigToExp.html) does what you need:

    4 (Cos[θ] - Cos[φ])^2 + (Sin[θ] - Sin[φ])^2 // TrigToExp

    (*    5 + 3/4 E^(-2 I θ) + 3/4 E^(2 I θ) - 3/2 E^(-I θ - I φ) -
          5/2 E^(I θ - I φ) - 5/2 E^(-I θ + I φ) - 3/2 E^(I θ + I φ) +
          3/4 E^(-2 I φ) + 3/4 E^(2 I φ)                                  *)

For more complicated functions without an explicit exponential structure, you can evaluate the Fourier integrals explicitly:

    f[θ_, φ_] = Log[4 (Cos[θ] - Cos[φ])^2 + (Sin[θ] - Sin[φ])^2 + 1];

    F[a_?NumericQ, b_?NumericQ] := 1/(2 π)^2 * NIntegrate[
        f[θ, φ] * E^(-I*{a, b}.{θ, φ}), {θ, 0, 2 π}, {φ, 0, 2 π}]

For example, sum up all Fourier components up to order 2:

    g[θ_, φ_] = Sum[F[a, b]*E^(I*{a, b}.{θ, φ}), {a, -2, 2}, {b, -2, 2}] // Chop

    (*    1.48242 +
          0.0545782 E^(-2 I θ) +
          0.0545782 E^(2 I θ) - 
          0.0291498 E^(I (-2 θ - 2 φ)) - 
          0.136866 E^(I (2 θ - 2 φ)) - 
          0.224188 E^(I (-θ - φ)) - 
          0.517258 E^(I (θ - φ)) +
          0.0545782 E^(-2 I φ) + 
          0.0545782 E^(2 I φ) -
          0.517258 E^(I (-θ + φ)) - 
          0.224188 E^(I (θ + φ)) - 
          0.136866 E^(I (-2 θ + 2 φ)) - 
          0.0291498 E^(I (2 θ + 2 φ))    *)

    Plot3D[{f[θ, φ], Re[g[θ, φ]]}, {θ, 0, 2 π}, {φ, 0, 2 π}]

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/YvabL.png