According to Merriam-Webster, "maturing" is "the process by which one becomes mature; develop fully". So it is the process of coming of age, of becoming an adult, losing one's innocence, etc. Basically...growing up! Bless Me, Ultima is centered entirely around Antonio's growth and maturation.
When the novel begins, Tony is only seven years old. He is happy with his family, but he understands there is a certain level of contention between his father's family, Marez, and his mother's, Luna. This struggle defines Tony's search for his own destiny. Will he be a vaquero, like the Marez clan? Or will he become a priest for the Lunas, to fulfill a prophecy? Seeking the answers to these questions becomes Tony's quest.
One example of Tony's growing maturity is his questioning the notion of faith and organized religion. Tony has been raised Catholic by his mother, but he has difficulty coming to terms with the vengeful, wrath-filled male God. He much prefers the kind, gentle Virgin Mary, yet cannot reconcile the two halves of the religion in his mind. His friendship with Florence solidifies this doubt. It is Cico's revelation of the golden carp that opens Tony to the possibility of other beliefs, and helps him to better understand his own faith.
A second example is Tony's acceptance of death. As a young boy, he experiences death in many forms. First, Lupito is shot in front of him at the river. He struggles with the horror of this tragedy, but Ultima helps him come to terms with what he has seen. Then, Narciso, who Tony turned to for guidance, is shot by Tenorio. Again, Ultima helps Tony understand the nature of his death. Ultima's support strengthens Tony, leaving him able to bear Florence's drowning, and eventually, Ultima's own death.