The problem is that a few of my colleagues and my boss are unwilling to use the ticketing system to report any bugs.
None of these address your boss also doesn't want to do this.
I would recommend three things.
First, talk to your boss and understand why your boss is unwilling to use the ticketing system. At the end of the day something is off here. You need to fill out a timecard accurately but your boss is actively working against your ability to do that - does your boss even know this?
If your colleagues are literally as you say "ranting" at you for five minutes and your boss doesn't care, your boss sucks. I personally suspect there is much more to this story than what you've presented here but regardless you need to talk with your boss.
Even questions like, "Bob doesn't want to submit bug reports, how do you want me to track that work?" is meaningful.
Second, do some serious reflection about the process your colleagues have to go through when reporting a bug. There are teams I work with that I have given up writing bug reports for because of how frustrating that process is.
I more or less have delegated the "is this a bug?" question to them as I have had way too many negative experiences working with them - whether it's closing the ticket as "won't do" or changing a bug report to a feature request or arguing with me about whether it's a bug, I only have patience for so much of that before I stop caring. I report the functionality to them via chat and let them decide if it's a bug. I have wasted far too much time trying to "prove" things to them and ultimately it's not my responsibility to make their product better.
Last, if the situation is actually what you are describing here, your workplace sounds really crappy. But - there are likely two sides to the story and what we're all missing (including you it seems) are the other story.
Empathy is important. Multiple coworkers, including your boss, "ranting and raving" rather than reporting a bug properly suggests there is a major lack of empathy somewhere in the interactions.
my boss [is] unwilling to use the ticketing system to report any bugs&&
Bugs need to be reported so I can put the time into my timesheet. Does your company require you to fill your timesheet? If so, does your boss not act by your company's standards?