Hi Zach,
An FM radio, even a cheap one, doesn't have only one or two tuned circuits for the signal:
1) Input tuned circuit. Maybe.
2) Collector load of input transistor. Probably.
3) 1st 10.7MHZ IF. Yes.
4) 2nd 10.7MHZ IF. Probably.
5) 3rd 10.7MHZ IF. Maybe.
6) 4th 10.7MHz IF. Probably not in a cheap radio. In a good radio, probably.
7) Discriminator, ratio detector or quad-coil. Yes.
Show us a schematic of the radio you want to modify and we will show how to increase the Q on each of the above circuits individually. It won't be easy and they would need to be tuned with special test equipment. The whole bunch will probably end-up oscillating.
He, he. My very 1st job (a long time ago) was repairing car-radios on the production line. The radios with the "best" IF transistors oscillated. Instead of replacing the transistors until I found one that worked OK, I just tacked-in a resistor across the IF transformer to reduce its Q and therefore the gain. I got paid a nice "quantity bonus" for that trick. I didn't let the other guys know about it, they just kept changing transistors all day long! I was also quickly promoted to engineering and design. ICs were just beginning to be made at that time. Lots of fun!
The guy at the end of the production line also got a "quantity bonus". There was a shaker there and he just scewed-in lightbulbs and tacked-in all the electrolytic caps that fell out (that was before hot-melt-glue and LEDs)!
If the radio uses ceramic filters for the IF then it is impossible to increase their Q.
A broadcast band FM radio needs a bandwidth of at least 150KHz:
1) Digikey sells ceramic filters for FM communications radios with a 3.75KHZ, 7.5KHz or 15KHz bandwidth. Way too low.
2) They also sell ceramic filters for FM broadcast band radios with a minimum bandwidth of 150KHz which the radio probably already has.
3) They also sell a tiny surface-mount ceramic quad-coil with a 500KHz bandwidth.
Do you still REEELY want to modify a cheap radio to be another "wonderful glorious super sensitive radio"?