How do I paint a gold framed mirror to look like stained brown wood?

This article on faux wood grain with paint might help:

https://www.hometalk.com/diy/paint/faux-wood-grain-paint-technique-27949293

Tape off the mirror section and get spray paint the color you want. However, I hope the gold is not real gold, but a gold paint.

Real gold? Really?

Paint it a tan or cream color, then go over that with a wood stain that has a polyurethane or varnish finish. I did this in my parents house on their wood work to make it low maintenance. It worked well.

I'd recommend using a brown glaze or water based wood stain. A few layers over the gold with a paint brush would get you the look you like!