They weren't actively trying for a baby, though they were open to it if it did happen. It wouldn't happen instantly (i.e. the first time they had unprotected sex) since there are some genetic differences between Humans and Vulcans. When Uhura found out, she was over the moon and wanted to tell Spock in a special way. She'd talk to Bones about how to tell Spock—since Bones would be the only person who knew. He'd give her various suggestions, all of which she'd ponder over immensely.
That evening, she and Spock would be having dinner together in their quarters. Uhura would tell him she had a surprise for him and would rush off to the bedroom to grab something. She'd return with a medium-sized box with a green ribbon on top. Spock would be curious and would ask her what the occasion was. She'd just grin and tell him to open it. Inside the box was a small, stuffed Sehlat—the Vulcan creature that most kids had as pets. Spock would take out the toy and look at Uhura with confusion. "Well, our kid can't have a real one as a pet, so I thought this would be the next best thing," she would tell him. "Our kid?" Spock would ask before his eyes went wide. "Nyota, are you...?" She'd nod eagerly and smile. Spock would put the toy down and stand up quickly to take Uhura into his arms.
They would both be so excited and would want to tell their friends and family, but would know to wait a little longer just to be safe. When they decided to tell their friends onboard the Enterprise, they would have them all over for dinner and would break the news after everyone was done eating. After that, they'd call friends and family who weren't onboard. Soon, their news would spread throughout the ship and they'd be receiving congratulations from various members of the crew.
Everyone on board would start to become overly protective over Uhura, even after she insists that she's fine and doesn't need special treatment. People would make sure to keep a close eye on her when on away missions and would offer to help her with anything and everything. She'd find it quite tiresome and would snap at a few people from time to time because she'd just feel so suffocated. Spock would gently explain to her that they were just worried something might happen to her or the baby, but he would promise to talk to the crew about not giving Uhura so much unnecessary attention.
Spock would want to make a thorough plan of how to raise his child in a way that wouldn't make them feel like someone with one foot in two worlds like he had felt. He'd express his worries to Uhura, who would assure him that they would give their child so much love and support that they'd never feel like an outsider. Though, growing up on such a diverse ship would help their child feel welcome and at home, and Spock and Uhura would make sure to teach them all about Human and Vulcan customs and cultures and would give them any support they needed.

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Star Trek Imagines (Character x Character)
FanfictionOne-shots and scenarios for different Star Trek couples (canon or not). I accept requests, but please check my bio on my account to see whether I'm taking more at a given time. Information about requests can be found in the first section of this. I...