What you're after is git hooks, specifically the post-recieve hook. look in your .git directory and there is a folder called hooks. Inside there are files that are basically shell scripts.

I would write a shell script to trigger your gulp command, test it, make sure it works, and then hook it up to your post-receive git hook like so...

!/bin/bash

path/to/your/shell-script.sh

Now whenever your repo on your server recieves changes, that shell script will run and do whatever needs to be done.

Here is a more thorough guide: https://www.digitalocean.com/community/tutorials/how-to-use-git-hooks-to-automate-development-and-deployment-tasks