Immortality is somewhat more complex in Arda than you may think. Yes, the elves are technically immortal, but in the sense that they do not die of disease and that they do not age when they go to Valinor. However, they can still be maimed, killed, and die of grief (in middle-earth).
Now, to more directly answer your question, elves have parents, grandparents, etc. The first elves at Cuiviénen had children, who would be younger than them, and so on. So elves did look at age, and there were age differences. Cirdan was one of the oldest elves in middle-earth, but for example Arwen was very young compared to him. As I understand, you are asking if elves look at age differences, and the answer is yes.
To give an analogy. You have an immortal being, but this immortal being had to be born. That being, while immortal, has not been around "forever", and so has an age. Now that age cannot be told from looking at that being, but that being still has an age.