No. The two species of fish need entirely different tank conditions. As others have said, a big temperature difference. And with Goldfish being so messy, if they caused a water quality upset the Betta would die first.
There's also the problem of feeding - the Betta would be barged out the way and its dinner stolen. Goldfish also fin nip and push other fish around. And an adult Goldfish could easily eat a small female Betta!
So the answer is no. Goldfish are best in large, over filtered Goldfish-only tanks.