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Do you know a place where the integral cohomology of $G_2$-homogeneous spaces is computed?

Great computational efforts using representation theory in order to determine the characteristic classes of homogeneous spaces were done by Borel and Hirzebruch in a series of papers:

  1. Characteristic classes and homogeneous spaces. I. Amer. J. Math. 80 1958, 491–504.
  2. Characteristic classes and homogeneous spaces. II. Amer. J. Math. 81 1959 315–382.
  3. Characteristic classes and homogeneous spaces. III. Amer. J. Math. 82 1960 458–538.

I was unable to find there a systematic answer to the question.

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    $\begingroup$ Can you clarify what you mean by $G_{2}$? For instance, if we're talking about the complex (equivalently compact) form, then the question has a complete answer, in the case of parabolic homogeneous spaces, in terms of Schubert-Bruhat cells. For the split form and it's parabolic homogeneous spaces, the question is more difficult, but there is still a lot of literature. This is reflected by user43326's answer below. In any case, for non-parabolic homogeneous spaces, an answer to your question entails a classification of sub-algebras, which exists, but the answer won't be so uniform. $\endgroup$
    – Andy Sanders
    Commented Apr 22, 2019 at 15:41

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I don't know how "systematic" the answer you are looking has to be, but for the quotients of the form $G_2/T$ and $G_2/P$ (P: Parabolic subgroup) you can find the results in Schubert presentation of the integral cohomology ring of the flag manifolds G/T, by Haibao Duan and Xuezhi Zhao

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