Can my 9 speed be converted to 13?

It's not one of the convertible models that could be changed with a shifter and air line re-plumbing if that's what you're thinking about.
The model you have would require a whole back housing assembly and a few other bits to change it over to a 13 speed.

 

It's gonna be some money and some parts

 

Not only do you need the housing but you need the extra gears that go in it and the air shift system to make it work…..Im guessing the only way you’re going to get the parts is by buying a complete transmission because once someone sells that stuff off a 13 speed they pretty much have scrap metal in the form of a 9 speed
I have been told some 9 speeds can’t be turned into a 13 speed this way because the gear ratios don’t work out….I’ve never explored it… just get a complete 13speed core of the specs you desire and go through it. it is possible to install just the back box with the transmission in the truck but in my experience it requires a lot of nice words and isn’t worth it
I typically pull the whole thing out if I’m going to have to remove the back box

 

Listen to this advice OP. It's not as easy as some make it seem, and you're either buying a core 13, or new parts.

If you can't do the install yourself, labor and downtime will eat up any savings over buying a trans and the labor to R&R.

Very few mechanics are skilled enough to even do this work anymore. And yes, the back box can be a PITA even on a 9 speed.

Add in the additional weight of the 13 spd aux countershafts and tapered bearings letting them lean like crazy, it's tough to get the aux section in place and also in time, even for experienced hands.

 

Unless I had a compatible aux section just laying around I'd replace the whole transmission.

You'll also need the driveshaft shortened because a 13 speed has more gears inside and therefore its longer.

 

Yeah, probably no one would give any core credit for it. I bought a Fuller 18 a couple years ago, and paid an extra 800$ i think in core for my 13 spd.