Unscrupulous shops or mechanics often target inexperienced/younger people for unnecessary work. Your coolant may indeed need to be changed but $120 to swap it is outrageous. I would look at the coolant yourself in the morning when the engine is cold & see what it looks like. Odds are good you can just look in the plastic coolant recovery container to see its condition.
If it's actually ok, don't feel bad about them trying to screw you. I was 21 with my wife driving across the desert one evening when I stopped for gas in a town called Baker Calif. which I now know to be notorious for ripping off customers to get business. They squirted some oil onto my shock absorbers & told me they were blown & that we'd have a wreck & die if we didn't let them change the shocks. I agreed since I was very protective of my wife. It was years later I figured out I had been scammed.
At the very least, take it to a small independent radiator shop & ask for their opinion. I would avoid the large chain garages, they are not as likely to be as honest as a small independent shop is. Especially if you can find one that is an active member of your local BBB (Better Business Bureau) which makes it far more likely they will be honest with you.