The most important to motivate your team is that you take the lead.

And your first action should be to stop smoking the carpet (talking nonsense).
No kitten needed : just a process to set-up, and set it mandatory.
Everybody has the habbit to registed time.
ProjeQtOr can be one of the way to do it.
But you call also use it to challenge team members.
For instance, you can extract resource who treated most tickets in the week : employee of the week

If you really have to motivate your team, you can also thing of a bonus...
But in the end, I shoud not convince you to use a PM Tool : if you are not convinced you need one and you much teach your team to use it, I cannoy help you much

I only can convince you that ProjeQtOr is the PM tool youu need