Question

Where does the T-lymphocyte matures once the thymus degenerates?

Answer 1

A few points, I'll expand

1. The thymus starts degenerating at puberty but continues to function in thymocyte maturation for decades, albeit at a lower level
2. Many mature T cells can live for years
3. Through homeostatic proliferation, peripheral T cells maintain their population size
4. Immune surveillance generally decreases with age

Now coming on to question
yes, the thymus involutes at puberty but that doesn't halt thymopoiesis. The thymus continues to produce naive T lymphocytes throughout our lives (even up to 70 years of age-see references) albeit at a very reduced rate. This mechanism is very important for example in HIV-infected individuals who are still able to repopulate their immune cells through a combination of peripheral T cell division and thymic production of naive T cells.

Note - Extended lifespans and homeostatic proliferation (points 2 and 3) don't contribute to TCR diversity but only maintain the existing repertoire.