It would not be right to raise a child in a very controlled environment where she is cut off from all that wonderful, colorful nature you get to experience every day. But we can think about such a child. So let's do this. Suppose this child is raised in such a very controlled environment where she can ever see only six basic colors: blue, white, black, red, yellow and violet. I think we would be able to explain to this child -- in fact, to any child -- that the color violet lies between red and blue, is really a mixture of red and blue. And I think we could then go farther and say that red and blue can also be mixed in different proportions, so that the mixture contains less blue and more red. In this way, I think we can get the child to imagine the color purple. So the answer to your question is yes, it is possible.
But it is possible only in those cases where the child knows colors that are close enough. You probably know from your experience in drawing that you can mix yellow, red and black to get brown. But I would be surprised if the child in our story could imagine brown with this instruction: imagine a mixture of black, red and yellow. Similarly, the color green can be produced by mixing yellow and blue, and again I think our little girl would not be able to imagine green from the instruction to imagine blue mixed with yellow. I am unsure about orange which, as you know, is a mixture of yellow and red. It would surely be harder for her to imagine than purple (where she already has another mixed color -- violet -- available to her). I am similarly unsure about turquois, which is a mixture of blue and green. Perhaps she would be able to imagine orange and turquois.
So the full answer is that which of the colors someone has never seen before she is able to imagine depends on the colors she has seen. New colors can be imagined if they are variations on, or combinations of, colors that one has seen before.