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:

- Characteristic classes and homogeneous spaces. I. Amer. J. Math. 80 1958, 491–504.
- Characteristic classes and homogeneous spaces. II. Amer. J. Math. 81 1959 315–382.
- 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.