No. The physics does not allow this. Physics does allow interstellar travel and colonization. However, the resources required are immense. To transport a single human being between star systems would require more energy than it would take to keep them alive in decadent luxury for centuries. We're talking hard limits of physics here. Sure, you can imagine this will be solved by some magical technology outside the known laws of physics, like a perpetual motion machine, but you could then just as easily imagine using that magical technology to keep people alive back at home.
Interstellar colonization would be a wonderful idea, but simply as a means of backing up life. And spreading life through the universe has value in and of itself. It's a wonderful thing to yearn for.
But interstellar travel has absolutely zero utility as a means of addressing population growth.
Oh, and you can only have so many humans living in any given volume of space before the raw output of their body heat cooks them all alive. Regardless of your tech, there are only so many humans you can cram into a volume the size of the solar system.