@Jamie ********
Sure. You have to get that extension of stay at the immigration office in your region. Each immigration office has some minor differences in the requirements, but generally they need:
1. The application form (you can get that at the office)
2. The marriage certificate and a front and back copy of the certificate.
3. Your wife's ID card, and a front and back copy of the ID card.
4. Your passport, and a copy of the data page of the passport (the one with the picture), the visa, and the most recent entry stamp,
5. 2 passport pictures
6. 1,900 THB
7. Your wife physically present.
But different immigration offices have different requirements, so ask and make sure.