Although it's commonly said that a nucleon contains three quarks, it's more accurate to say that the net number of quarks (i.e. number quarks - number anti-quarks) is equal to three. The interior of a nucleon is a fantastically dynamic place and the number of quarks and gluons at any instant is typically much greater than three.
So it is wrong to imagine the proton as containing a red up, a blue up and green down quarks. All you can say is that the net red charge, net green charge and net blue charge will all be one.
You comment gluons will be emitted to change colors so annihilation happens is sort of true, but it's a bit misleading to imagine there is, for example, a distinct down quark that is green and you need to wait for a gluon to be emitted before it changes colour. A better way of describing this would be to say the down quark is in a superposition of all colour charges.