There was no plumbing ran to the third floor of our home. We would like to add a bathroom there to make a loft apartment for our grown son. Can we do this?

Yes. Likely depending on where you live. This would require a building permit in most locations, where are you living? Are you considering a full bath, 3/4 or half?

Issues of where the existing plumbing is, and where you want the bath. Water volume and pressure currently in house.

But normally yes you can. As noted you will have to get a building permit, have the new installation inspected. Normally the area above where the existing water and drain pipes are located work best.

Is the floor able to support this potential load? Many 3rd floor spaces were designed as attics and need floor joist replacement or sistering. A steel tub weighs 100# and then 300# or so of water. Add the toilet, vanity and you have a fairly concentrated load in a typically small footprint. You will need water supply, drains and a vent system. Make sure whoever does the work looks at the floor underpinning.

Ma0641 said it: check your ceiling joists. Usually they are undersized as they are not intended to carry any load. You would have to reinforce them / or double them up / with right size members. As far as plumbing goes, yes, it can be done

Milo