Depends on the "profile" of the Qualcomm. The "profile" is the instructions that tell the QC how to behave under certain conditions, like when the ignition key is on or off. Most QC's are set with a profile that will have it go into QUIET mode a few minutes after the ignition is turned off. Look at the QC screen, STATUS: ????? and it will tell you GOOD or QUIET.
If the ignition is off, most of them will stay in QUIET mode for an hour or two, then go to sleep, shutting off completely. Then, every hour or so they will wake up, check for messages, then go back to sleep after 5 minutes. After 6 or 8 hours of ignition-off condition, the QC will no longer wake up - it just sleeps forever until you turn on the ignition again.
That's the normal, most used, standard profile for the QC, and it is designed to not run down a battery. If your QC is remaining on, or shows GOOD status after the ignition is turned off, then, yeah, it will, absolutely, run the battery down overnight.
If your QC has the normal profile, and your replacement battery drains, then it's not an old, worn out battery. Assuming the alternator is working properly and is charging the battery, then you have something drawing a load when the ignition is turned off.
Any mechanic or anyone with a ammeter can tell you in 30 seconds what your ignition-off no-load draw is on the battery. If you have a no-load draw, you'll need to find out what it is. If you have an inverter hooked up to the main battery instead of an isolated aux battery, for example, some of those have a high enough no-load draw to run down a battery. An XM Radio receiver plugged into the cigarette lighter adapter can run down a battery if left on for a couple of days.