Can I notate time complexity depending on the result of an algorithm? I have a program that calculates the first year some two events happen on the same day. I have not calculated any upper limit for the year. The time complexity of the program would be equivalent to O(Years), where Years is the result of the algorithm. 

Is such a notation possible?

If yes, can I call the algorithm "linear"?

If not, what would be the correct way of notating the complexity?
You can express algorithms costs in whichever parameters you desire, so long as you define them. The parameters don't even have to be known given the input; that gives your bound limited use, but well.

Warning: Landau bounds are asymptotic, that is they only hold meaning when you let the parameter(s) go towards infinity. That is you can not really use parameters that can not be arbitrarily large.

Warning: It's unlikely that your cost is independent of the input, which is what your bounds seems to suggest. You probably made a mistake.