Was the Zimmerman telegram a forgery?

No, it was a genuine attempt by the German government to persuade Mexico to go to war with the US, in the hope of keeping the US so busy in its own backyard that it would be unable to get a large army to France to fight the Germans. The Germans had just taken the step of resuming unrestricted submarine warfare, which they knew was quite likely to, and in fact did, precipitate the entry of the US into the war against Germany. The Zimmerman Telegram revealed the Germans' effort to avoid the worst effects of US entry into the war by diverting US efforts into a war with Mexico.