Most jewelry stores used to have actual jewelers working in them and could do that. However, some shops today, do not have actual jewelers there. Those are simply retail stores with sales staff and send out their repair work. I used to get repair work from a couple such jewelry stores in the mall.
Also, if there is a rock shop or lapidary shop near you, most likely they could do that.
Frankly, anyone with a diamond bit for their drill could drill the hole.
One last note, rose quartz is a very inexpensive (in a jeweler's sense) semiprecious stone. The cost of drilling the hole will probably exceed the value of the stone. It may be more practical to buy a drilled rose quartz bead. However, if this particular stone has special meaning to you, then the cost is not the primary factor.