Is there a fuse for the fuel injector for a 1989 buick centry?

Yes, but I don't know which it is. Here's how to test all under hood and under dash fuses:

https://www.askmehelpdesk.com/cars-trucks/faq-how-troubleshoot-repair-maintain-hondas-selected-other-vehicles-46563-3.html#post252145

Fuel injectors are very reliable. If your car doesn't start, the odds are 90%+ that it's electrical related, not fuel. Prime candidates are Ignition Control Modules (ICMs) and coil packs.

BE VERY CAREFUL HERE!
With the key in the run position, check for continuous power on the PNK/WHT wire at the injector.
The fuel injectors ground side is controlled (pulsed) by the Engine Control Module (ECM)
If your not getting injector pulse, you should verify that the ECM "sees" ignition pulse/engine RPM.