I don’t know that “utopia” is a discrete enough idea to quantify something like birthrate anyway.
Here’s a Utopia: Very industrialized nation where everyone works about 4 hours a day doing something they have been trained and selected to do based on their personal preferences. Every person is employed, and most people work inside, by themselves, either with computers, remote systems, or mechanized helpers. Hobbies are pursued almost religiously - excrcise, sports clubs, creative arts, all of these provide socialization for people outside work. Individuality, creative output, and personal autonomy are highly prized, and personal accomplishments (through work or play) are considered the best way of quantifying a person’s worth. Marriage occurs, but isn’t considered socially necessary, more of an organic extension of two highly-compatible “soul-mates” who have decided to join their lives.
Here’s another one: Agrarian society where advances in robotic and mechanized labor have made crop yields stable, and physical labor not as draining. However, the established religion focuses on remaining close to the earth and personally overseeing work, so most people are farmers or livestock keepers. In addition, the idea of community is valued above individuality, and community-members learn early where their talents lie in order to better support the whole. The focus on growing crops and tending to the community make large family blocks highly desirable, and entire clans develop among the individual crop-growers. Marriages are highly socially and religiously regulated (that focus on cultivating and growing again), and alliances between clans and families by hosting “foster children” or “clan wives” are sought after by everyone. Family dynasties control the various crops and goods produced.
So, you’ve got two societies, nothing like each other, which could both be utopian in nature for a good number and type of humans to flourish. The first one is going to die out, and the second one will grow (but probably split off schisms) rapidly.
Can you think of why that is? 
Basically, if you want to create a ficticious utopian community with a particular birthrate pattern, you have to figure out what social pressures will contribute to that. Alternatively, if the social structure is more important to you, then think about what the impact of those structures and ideologies will be for birth rates after you establish what you want to have.