Okay that appears to be UBBThreads.
So yes you would upgrade to 6.5.5 then at least two more upgrades.
You would upgrade to ubbthreads 6.5.5 then import to like 7.57 or 7.60 before upgrading to the current version.

the caveats are.
One around ver 7.2 global moderators were introduced which sometimes causes a issue with moderators.being placed in the wrong group
Not really hard to fix.

The other is kinda big that is you need to downgrade your php version to like 5.4 to 5.6 till you get to the UBB7.7.X then you can change to php7.x
If your host does not allow that then you would need to do it on a local host then upload. Which you probably are already on say php 5.3 or 5,4 at your current host .


I think if you go through the upgrade you will find that if you look you will find many enhancements and bug fixes that are not in your current version and may decide to stay with UBB,