So I'm dm ing for the first time, and my group captured a goblin who I am hoping they will be able to turn to their side.

That said, should the goblin be involved in any battles with them, is there any mechanic for that goblin to gain levels?

He won't be going on every adventure but if the group are waylaid while traveling to their base I feel he should get some progression if he helps.

Depends on what edition you're playing (and what game, technically, but since you say "DM" I assume this is some version of D&D).

If it's 3rd ed (or Pathfinder, I assume), and the goblin has default stats, then it's assumed to be a level 1 Warrior, as described in the DMG.  In that case it can just gain a share of xp like any other character and level up accordingly.

If it's 2nd ed then there's no official mechanic (unless it's in one of the other books that I never got).  You just increase his stats in whatever way makes sense to you, whenever you think you should.

If it's any other edition then you'll have to wait for someone else to come along and answer...

5th edition d&d obviously I have the stats from the mm, and I guess one obvious jump would be to turn him into a goblin boss after a couple of levels. But I was wondering if I could take that further.

You should be able to take it as far as you'd like. I've got a current goblin DMPC who runs with the PC on a solo adventure. Currently he's a 4th level rogue. You can tack it up really however you'd like with 5th's flexibility. But I commonly will count their monster level as typically some sort of level 0, give them some small amount (depending on their CR) of XP needed to be able to take an actual class if desired. Otherwise, you should be able to keep it as the monster type and bump it to a goblin boss, and then just add on whatever you feel would work for the goblin (boss), the party, and/or the campaign.

The only limit is your imagination.