It would help to clarify the context of "Chevalley group" here, but over any field one gets uniform results by starting with a Chevalley basis for the associated complex Lie algebra. The commutation relations for rank 2 groups including $G_2$ are given explicitly in SGA3 and written down similarly in my 1975 Springer text Linear Algebraic Groups in section 33.5.

It would help to clarify the context of "Chevalley group" here, but over any field one gets uniform results by starting with a Chevalley basis for the associated complex Lie algebra. The commutation relations for rank 2 groups including $G_2$ are given explicitly in SGA3 and written down similarly in my 1975 Springer text Linear Algebraic Groups in section 33.5.

ADDED: There are two cases not involving a pair of positive roots. 1) If the two roots are independent, they both lie in some positive system. Using the standard picture of the 12 roots, it's easy to visualize the Weyl group transformation involved. Then just relabel the two roots by the corresponding two in the fixed positive system and apply the formulas worked out there. 2) If the roots are negatives of each other, life gets a little more complicated. But here the commutation takes place in a rank 1 group. Explicit commutation formulas here are usually avoided in the literature but take place just in a group like $\rm{SL}_2$.

Maybe it would also help to add two relevant online references. Both are based ultimately on a Chevalley basis (keeping in mind that such a basis is unique only up to certain sign choices) as well as the ideas in Chevalley's 1956-58 classification seminar.

SGA3 is now online here. See in particular Expose XXIII.

Steinberg's 1967-68 Yale lectures on Chevalley groups are available at his homepage here. See Section 6, page 66.