There's nothing special about going to Ab instead of Fm. Maybe there's a name for it, but I'm not good with names, it's easier to think of progressions in terms of what they do and what chords they consist of.
I see three things about the chords you ask about
- During the first four chords, there's a descending chromatic motion Eb - D - Db - C. You could emphasize that by making the chords Bb/D and Bbm/Db, with D and Db in the bass: Eb - Bb/D - Bbm/Db - C7. (Edit. Ok, seeing John Belzaguy's answer, that's what the song actually does)
- The Bbm - C7 chords make you think it's going to do a secondary dominant motion to Fm. When you're in Eb or Cm, the chords Bbm - C7 (or Bbm6 - C7 or Gm7-5 - C7) are commonly used as a two-phase secondary dominant springboard to Fm. You could use just C7, but preceding it with Bbm (or Gm7-5) is an additional booster.
- If you prepare a dominant-tonic motion to a minor tonic, you can instead opt to go to the relative major tonic instead, and then it will just sound like "ah it went to the relative major instead". Here's an example in Fm or Ab that seems like it's going to Fm, but it goes to Ab instead.
You can do that, and it sounds like what it sounds like.
In the example progression you listed (I didn't listen to the song), the things are combined - a secondary dominant motion to a minor, but going to its relative major instead. It seems like going to Fm, but unexpectedly it goes to Ab. And it does this so that Eb feels like the real tonic. How unexpected this really was, it depends on what you expect. Maybe by the time you get to the C7 you're like "OMG it's not going to do this cheezy grandpa music jump!!? I have to press stop quickly before my friends hear this, because it's going to be soooooo embarrassing if it goes to Fm." And then you're relieved to get the Ab instead, retaining at least some level of credibility among your social group.
There are so many different ways to combine chords, if you wanted to assign a special name to every useful trick, you'd need to learn a lot of special names to know what you're talking about. Instead you can list the chords.