Yes (sort of).  It would seem that maybe Black could try to make use of not having committed his bishop to g7 (against the e5-plus-f4 system he would often later redeploy it via f8, thus costing two tempi; that's part of the raison d'être of the above-mentioned "Accelerated Gurg[enidze]," 1. e4 g6 2. d4 d6 3. Nc3 c6 4. f4 d5), but it's not clear to me how.  So it may seem that after 1. e4 c6 2. d4 d5 3. Nc3 g6 4. e5, Black wouldn't want to play 4...Bg7, and one might think that he wouldn't want to play ...h5 until White has played f4.  I might wonder if 4...Qb6 could have some sense (perhaps meeting 5. f4 with ...h5 and 5. Nf3 with ...Bg4); ECO only mentions 4...Bg7 and 4...h5, thinking each should lead to += (and indeed, on 4...Bg7 White plays 5. f4, while on 4...h5, he plays without it).

If you are going to play in non contact style with c6/d6/d5/g6 and Gurg whatever, it doesn't matter much from Black's perspective what order you play them. You can throw in a Bg7 at some point too, unless you want to do an "Accelerated Gurg"

So the choice depends on your oveall Black repertoire. For a Modern player, g6 first maintains most flexibility. Though the d pawn often goes to d6, it helps to keep some flexibility for d5 if you want the Gurg to be part of your repetoire. For the c file, c6 is a commitment as Nc6 and c5 options are now for the most part ruled out.

   

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Well, there are the obvious deviations like 3.exd5 and 3.f3, but that is not your question I suppose.
1.e4 c6 2.d4 d5 3.Nc3 g6 4.f3 Bg7 is not such a good verion of the 3.f3 variation.
Overall I think White can prove a sound += against the Gurgenidze, both with 4.Nf3 Bg7 5.h3 and 4.e5 Bg7 5.f4. The key in the latter is to start a queenside attack with b3 and c4.
An advantage of the C-K move order might be, that Black can delay ...Bg7, eg 4.e5 h5 5.f4 Bg4. It has been long since I have looked at this.

    How do you like this position  :     4.Qf3 dxe4 5.Ne4 Bg7 6.Bc4 Nf6 7.Ne2 Nd7 8.h4
and White has other (attractive) alternatives to e5 such as the simple and efficient 4.f3.