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I would like to know if you have any reference where I can find the canonical PBW basis for $U_q(\mathfrak{g}_2),$ computed using the action of the braid group as defined by Luzstig.

Alternatively I would like to know if I should expect Luztig's root vectors to be given by some sort of $q$-commutators or linear combinations of $q$-commutators.

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The canonical basis and the PBW basis are two different things. I don't expect a simple computation of the canonical basis (in G2 the appropriate cluster algebra is not of finite type) while the PBW basis is not hard to compute for G2 as there are only six positive roots.

Each of Lusztig's root vectors for the non-simple roots are scalar multiples of q-commutators of appropriately chosen other root vectors. This dates back to Levendorskĭ-Soibelman (Proposition 5.5.2), but the only place I know of where the scalar multiples are determined is in my paper with Brundan and Kleshchev (Theorem 4.2). The results mentioned in this paragraph are true in all finite types, not just G2.

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