Yes, technically. but the trickle charger won't be doing anything.
Depending on the trickle charger, it might be able to sense the voltage, and while everything is running, the system voltage should be around 13.8 volts, and the trickle won't actually be doing anything.
Craig T. Bailey
2015 Jayflight 32BHDS
2007 Chevy 2500HD Crew 4x4 Z71