Is there a way to change the environment variables of another process in Unix?

Substantially, no. If you had sufficient privileges (root, or thereabouts) and poked around /dev/kmem (kernel memory), and you made changes to the process's environment, and if the process actually re-referenced the environment variable afterwards (that is, the process had not already taken a copy of the env var and was not using just that copy), then maybe, if you were lucky and clever, and the wind was blowing in the right direction, and the phase of the moon was correct, perhaps, you might achieve something.

You probably can do it technically (see other answers), but it might not help you.

Most programs will expect that env vars cannot be changed from the outside after startup, hence most will probably just read the vars they are interested in at startup and initialize based on that. So changing them afterwards will not make a difference, since the program will never re-read them.

If you posted this as a concrete problem, you should probably take a different approach. If it was just out of curiosity: Nice question :-).

Via gdb:

(gdb) attach process_id

(gdb) call putenv ("env_var_name=env_var_value")

(gdb) detach

This is quite a nasty hack and should only be done in the context of a debugging scenario, of course.