Generally, no. Most states with an income tax offer a credit for a tax paid to another state. So you will file a Georgia tax return to report the profit on the GA property, and pay the appropriate tax to GA. Then prepare your CO return, reporting the GA property income. CO will then give you a credit that reduces your CO tax for the amount of tax paid to GA, up to the max that would have been due if it was taxed only in CO. Of course, if GA taxes less than CO, you may have to pay additional CO tax up to the max tax due in CO.
If you bought in a state with no income tax (say, Texas), then CO would grant no credit, but would tax the full amount.
There are a few states with a somewhat more complicated arrangement where you have to go back to the state to which you paid the tax to get a credit. Finally there are some states with what can be called paired-state reciprocity. Basically, if the two states have a reciprocal agreement, and you live in State A and work in state B, then you would simply have tax withheld and paid to state A.