The short answer for a would-be talker
No, a zebra finch cannot learn to talk. Talking in birds — mimicking human words — belongs to parrots, mynahs, and a few other groups whose brains and vocal learning work differently. A zebra finch will never greet you by name, copy a phrase, or pick up words from the television, no matter how often you repeat them beside the cage. If spoken words are the main thing you want from a bird, a zebra finch is the wrong species, and it is far kinder to learn that now than after you have brought a pair home.
What a zebra finch actually learns
What a male zebra finch does learn is song, and he learns it early and seriously. A young male listens to his father, or another adult male nearby, during a sensitive period in his first months of life. He practises in private, starting with soft, messy subsong, then shapes it into a finished pattern of beeps, trills, and buzzy notes unique to him. Once that song crystallises, it barely changes for the rest of his life. Females never sing, but both sexes use quiet contact calls to keep in touch across a cage or aviary.
Why the distinction matters when you are choosing
Zebra finches are flock birds sold cheaply and often bought on impulse, sometimes with a vague hope they will be entertaining talkers. Their real appeal is different: constant gentle motion, soft social chatter, males singing through the day, and pairs raising chicks readily. They are watch-and-listen birds, not handle-and-chat birds, and most do not enjoy being held at all. A buyer who understands that tends to house them properly — in a wide flight cage with company of their own kind — rather than in a small cage alone, waiting for words that will never come. They also do best in pairs or small groups, because a lone zebra finch is a stressed zebra finch: the contact calls and answering songs are social glue, not decoration, and a single bird deprived of them often goes quiet in a way beginners misread as contentment.
If talking is what you truly want
Be honest about your priority before you buy anything. Species that commonly mimic include budgies, who can build surprising vocabularies in a small body, and larger parrots such as African greys, who bring decades of commitment, noise, and need along with their talent. Each step up in talking ability is also a step up in time, mess, volume, and cost. Many people who compare the two honestly discover they prefer the finch's undemanding company after all — but that should be a deliberate choice, not a misunderstanding discovered at home.
Settling in with a singer instead
Give a new male finch a few quiet days, then listen. Note which calls appear at feeding, which greet you in the morning, and how his song changes when another male answers from across the room. That listening is the real relationship this species offers, and it deepens for years. When an unfamiliar call puzzles you, put a follow-up question to KinsLe's Kimmy; she answers using the KinsLe Answers library.