Floor joist support

I think I would "sister" another 2x10 in the area the fireplace is going to be sitting. Might even do a couple of crossbraces too. Truthfully, it's pretty difficult to answer your question without knowing what else is in the room weighing on the floor.

The OP is adding a permanent load, which is a dead load. The duration of the load affects the calculation somewhat, and the allowable live loads in most span tables are not based on permanent loads. They're down-rated like wind and snow loads, though not to the same degree.

So, he must have to look at the tables for "dead loads" I take it. I assume they have those, unless that would have to be an engineers call.

The OP is adding a permanent load, which is a dead load. The duration of the load affects the calculation somewhat, and the allowable live loads in most span tables are not based on permanent loads. They're down-rated like wind and snow loads, though not to the same degree.

The tables show only distributed loads, not point loads. Point loads require calculation and won't be in any table. Not only ultimate strength will have to be determined, with a factor of safety of course, but shear and deflection. The deflection won't be a problem most likely. Having the load near the wall minimizes the risk, but if the guy wants to look into it he should do it the right way. That link gives a good overview of how to do it.

OP-

What is the TOTAL weight of the entire fireplace installation? Framing, fireplace insert & all the finishes (rock face)?

If the load is in the 600 lb range .....13 ft span at 40 psf is 520 but you have more like a concentrated load (really a local distributied load)

Just a SWAG but one sistered joist wouldn't be enough, two would probably be ok, three would definitely do it. But to be sure you need to do ( or have done) some calcs. That means....get it looked at.

I'd open up the floor or the ceiling below & sister in the joists.

cheers Bob