Our company doesn't have any problems

Liars!

Does absolutely everything run perfectly all the time?

If it does, then you are probably radically over staffed, which itself is a problem (for the shareholders at least.)

Regardless, problems occur for many reasons. They are opportunities for improvement. And should be worked on as a part of normal operations, not as a one off, "Let's get this handled so we can go back to normal."

Unfortunately, just staying at 'Normal' means you are losing ground to your competitors, who are constantly improving.

Problem Solving is not a specialized Art. Everyone does it to some extent. Often it is haphazardly applied, based on the skill and experience of the individual problem solver. When a problem is solved to root cause, it is rarely communicated outside of the group who solved it.

Problem solving should be just another system (like the dreaded accounting system) so that problems are solved systematically and can be communicated / referenced by the whole company. 8D (Eight Disciplines) is a thorough process. So is Toyota's A3. Your company might have your own variant of one of these processes. As long as you have a thorough, consistently applied process, you are good.

If you are a small company, a problem solving binder will do. A little bigger and you'll need a file drawer. Everyone needs to be able to enter and solve problems, so manual processes will do when everyone can get to them. As soon as multiple binders or filing cabinets appear, you'll need an automated system, since adding paper to files and then leafing through the information is unlikely to happen unless you have a librarian position watching over the archive.

Now find the time and start problem solving as part of the normal process. You'll be amazed at how fast things improve.