No, bettas and goldfish are not compatible for a few reasons.
First, as was mentioned, bettas are tropical fish and goldfish are not. This means that a betta needs to have water around 76-80ish degrees (everyone has different opinions on this) and goldfish do best in colder waters.
Another reason is space. Goldfish are pond fish. Contrary to popular belief goldfish don't belong in bowls because they grow to be HUGE. The reason why goldfish die so quickly is because they are put in a bowl, which is too small for them. Bettas should be in a reasonable amount of space, five gallons or more, but they don't need to be, and shouldn't be, in a pond.
Third, goldfish produce a LOT of waste. That waste builds up ammonia very quickly, which can destroy a betta's fins. Even if the other differences were aside, this problem would still keep them apart. The bioload can be difficult to keep up with and it could potentially kill your betta.
Also, if you were to get a fancy goldfish with the flowing fins, a betta would fight with it because it its fins.