I can see why you might expect Qxg4, but given that I'm the one down material, I'm not as inclined towards swapping...
1. g3 d5
2. Bh3 Bxh3
3. Nxh3 Nf6
4. Nf4 e5
5. 0-0 exf4
6. gxf4 Nc6
7. e3 Bd6
8. b3 Qd7
9. Nc3 0-0-0
10. Qf3 Ne7
11. a4 Qg4+
12. Qg3
1. g3 d5
2. Bh3 Bxh3
3. Nxh3 Nf6
4. Nf4 e5
5. 0-0 exf4
6. gxf4 Nc6
7. e3 Bd6
8. b3 Qd7
9. Nc3 0-0-0
10. Qf3 Ne7
11. a4 Qg4+
12. Qg3