You can't. The handling of shell jobs (foreground and background processes) is done by each individual shell instance. You can't migrate a job from one interactive shell to another, not even within something like screen.
The terminal doesn't really have anything to do with this. It just runs the shell.
What you may do is to migrate a whole shell session with a terminal multiplexer, such as tmux or screen. A shell session within one of these may be moved to another terminal through detaching the screen or tmux session in one terminal and reattaching to it in another on the same machine.
With tmux this is done by first starting tmux, then starting your program inside of the tmux session. When the program is running, you may press prefixd (prefix is Ctrl+B by default) to detach the tmux session. In another terminal, you then give the command tmux attach to get the tmux session back. Detaching can also be done through the command tmux detach.
See the manuals for screen and tmux on your system.