No, you should not deep discharge UPS without good reason.
At least not the overwhelming majority of small UPS devices, including those sold by APC. They are generally using lead–acid batteries, neither designed nor expected to endure many full discharge-recharge cycles.
Such batteries are not significantly affected by memory effect performance degradation that is said to occur in various consumer electronics. The (commonly misattributed into the same category) effects of over-charging or merely approaching their maximum recharge cycles are not relevant for UPS either, as the charging mechanism are already configured to maximize the batteries utility over the respective promised lifetime.
The only reason you would ever want to deep discharge an UPS is to test whether your setup really works as expected. Even then, do not overdo it:
Only perform runtime calibrations on your UPS one or two times a year,
if necessary. Some of our customers want to check their systems to
verify that their runtime is sufficient. However, consistently
performing these calibrations can significantly decrease the life
expectancy of your APC battery. -- APC FAQ 158934
If you do such, check the manual for your specific device - it may recommend that for testing purposes, you only use the devices self-test program, collecting data about the batteries remaining quality in order to correctly perform functions such as "send shutdown signal to attached machines when remaining energy lasts for X minutes".