Hi I had a similar problem, took it to main Fiat Agent they quoted £1300 to replace the front air con rad (condenser) they said it is common problem as the odd stone can damage it. If you look through the grill you may see lots of fly debris . Rather than trust Fiat agent I took the car to a air con specialist in Milton Keynes I have copied my post below, it may help.
Just out for what they can get out of you!
My Air conditioning is now blowing cold, I took the car into Glyn Hopkins Fiat Milton Keynes hoping they would correct the air con under my extended warranty NO! I left the car with them for the morning & when I collected it they presented me with a bill for £168 for doing a health check on the car. Did they do the job I asked ? NO ! “Sorry your AIR CON is not under our warranty”. 40 miles & 3 weeks prior to letting them have my car I took it to my non Fiat garage for an M.O.T it passed with no problems also a report on brakes & pads which were fine, however Glyn Hopkins in Milton Keynes said I need new disks, new pads, a new air con condenser, new timing tensioners all this they wanted £3000 The brakes they said needed immediate attention, they had the service history with the car that clearly shows everything is up to date & in good order. Not surprising I told them to get lost. I then took the car to G.P Auto’s in Milton Keynes they re-gassed the air con & also added a leak stop fluid as they said the gas leak was just a tiny pin prick of a hole it should do the job with no problem. Yes it has, so the moral is BE VERY CAREFUL what MAIN DEALERS TELL YOU! 2 weeks later & still blowing nice cold air.
The total bill from a NON DEALER £128!
Slight but important info- According to Fiat Dealer List Glyn Hopkins are not on their list of dealers, they are only listed as Assistance not retailers in Milton Keynes, the nearest Dealer is Motor-Vogue in Bedford
If it is the condenser where a stone may have penetrated it and the hole is small enough its well worth going to an air con garage who may be able to sel it then re gas the air con. It worked for me and is still working 2 years later. Mine is an Abarth Anniverario