Yes, it's easy to replace just the seat bottom upholstery. Once the seat is out of the car (four bolts), you will need to clip off the hog rings that secure the upholstery to the bottom, and take out the tension bar on the front and back edges by bowing it out in the middle. Then the thing comes off easily, and you don't have to remove the seat back. Be careful of the seat heater. You'll need some new hog rings and hog ring pliers to get the new one on, and you'll of course want to transfer all the tension wires and bars onto the new piece of upholstery.

Your biggest problem is where to find the new seat bottom upholstery without costing an arm and a leg. I've occasionally seen leather upholstery in the junkyards, but invariably the driver's seat at least is a bit torn up. I don't even want to imagine how much Volvo would charge for it. Do you have any equity in your house you can borrow against?