No. The trinity of persons cannot be known by natural reason because man cannot know and with his understanding grasp that for which no necessary reason can be given.
Man cannot obtain the knowledge of God by natural reason except from creatures. Now creatures lead us to the knowledge of God, as effects do to their cause. Accordingly, by natural reason we can know of God that only which of necessity belongs to Him as the principle of things.
The creative power of God is common to the whole Trinity; and hence it belongs to the unity of the essence, and not to the distinction of the persons. Therefore, by natural reason we can know what belongs to the unity of the essence, but not what belongs to the distinction of the persons.
We must not attempt to prove what is of faith, except by authority alone, to those who receive the authority; while as regards others it suffices to prove that what faith teaches is not impossible.
RO2: Reason may be employed in two ways to establish a point: firstly, for the purpose of furnishing sufficient proof of some principle, as in natural science, where sufficient proof can be brought to show that the movement of the heavens is always of uniform velocity. Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results, as in astrology the theory of eccentrics and epicycles is considered as established, because thereby the sensible appearances of the heavenly movements can be explained; not, however, as if this proof were sufficient, forasmuch as some other theory might explain them. In the first way, we can prove that God is one; and the like. In the second way, reasons avail to prove the Trinity; as, when assumed to be true, such reasons confirm it. We must not, however, think that the trinity of persons is adequately proved by such reasons. This becomes evident when we consider each point; for the infinite goodness of God is manifested also in creation, because to produce from nothing is an act of infinite power. For if God communicates Himself by His infinite goodness, it is not necessary that an infinite effect should proceed from God: but that according to its own mode and capacity it should receive the divine goodness. Likewise, when it is said that joyous possession of good requires partnership, this holds in the case of one not having perfect goodness: hence it needs to share some other's good, in order to have the goodness of complete happiness. Nor is the image in our mind an adequate proof in the case of God, forasmuch as the intellect is not in God and ourselves univocally. Hence, Augustine says (Tract. xxvii. in Joan.) that by faith we arrive at knowledge, and not conversely.