General:
* Sprout
* Bud
* Shoot
* Develop
* Grow
* Emerge
More specific:
* Seed (for seeds specifically)
* Incubate (for the process of development, especially in eggs)
* Take root (implies the process of establishing and growing)
* Flourish (implies successful and vigorous growth)
The best synonym for "germinate" will depend on the context of your writing.