In a Time Series Problem, is it possible to forecast quantities by learning the patterns of other items? What are my options?
If I am only concerned about Red Apples, I can just use ARIMAX with data pertaining to only red apples or VAR right?

Yes, is possible, you will lose information (perhaps valuable) but you can work with it.

But what if I suddenly introduce a new apple type such as orange apples and only have a history from T1-4?

Given I need to use T1-4 for the orange apples... Is it possible to use other data from the red, green and yellow to aid in properly calculating a QN value for orange apples at T5?

You will be limited to the knowledge of the orange apples but the other variables might help you obtain enough information until you could have a more robust model, 4 entries might not be enough, but the model might help you obtain an estimation