Can I extend my InDesign Server trial license?

Unfortunately, Typefi cannot renew or extend an Adobe InDesign Server trial license beyond 90 days, under any circumstances.

Adobe only allows one trial license period per version upgrade. You can see more details on this post by Mike Zahorik, Senior Business Development Manager, InDesign and Illustrator, Adobe. This is mainly aimed at Adobe partners and it states (emphasis added):

Customers that are transitioning from an existing version to a new version may continue to run the old version for up to 90 days while they install and test the new version.

For this reason, we cannot renew or extend the 90-day testing license, even if it is installed on a different machine. 

If more time is needed for testing, you can procure a development (non-production) license instead. Please contact your Solutions Consultant for more information.