First, consider that we exist in a Utopian society, no wars and no disease. A result of this Utopian society is that our technology has advanced considerably, improving healthcare and life expectancy (which averages around 100 years). Nobody is born with disabilities, there are zero cot-deaths, miscarriages or similar issues. Everyone is born healthy and is guaranteed to have a long, healthy life until they are roughly a century old.
Our advanced technology also gives us the means with which we can travel the immense distances of space and colonize other worlds. We have some form of FTL, so the travel time to these new worlds is measured in days or weeks. The worlds we colonize are effectively identical to Earth, and none of these worlds pose a threat to our survival (in terms of hostile native life or similar threats). This gives us all the land we need to support a population of 1 trillion. New worlds can only be colonized by a minimum of roughly 10,000 people. To keep administration simple, we can only colonize X worlds at one time, where X is the number of worlds that have at least 5 billion people living on them (so to start with we can only colonize 1 world, but once we have colonized that one, we can then colonize 2 worlds, and then after that 4 world and so on).
In this Utopian society, population growth rates are effectively governed by available space. The population on a world will no longer grow when it reaches 10 billion people (eg 2ish children per couple - whatever required to maintain the population but not exceed it). However a newly colonized world will frequently have birth rates within the 5-10 range - and this birth rate will slowly dwindle until the world has 10 billion people, at which point the population is fairly self sustaining.
Given these "perfect" conditions, and ignoring any possible interference from aliens or events that destroy entire worlds, how long would it take for 7 billion people to reach 1 trillion people.
Some additional details:
Consider this Utopian society to be very similar to a developed nation here on Earth for all concerns other than what I have described above; so similar gestation times, similar puberty ages etc.
Roughly a 50/50% male to female split for the population. People in this Utopian society wait until they are in their mid 20s before having kids, and consider a roughly 1-2% of people die from unnatural causes (inc suicides and accidents) and roughly 5% of people decide not to have any children.