I'm thinking of making a car battery with 3.7 V Li-Ion batteries arranged in 4s configuration (14.8 V). Would an alternator that produces 13-14 Vdc be able to charge the battery pack at all?
The alternator itself will produce much higher than the car's operating voltage - vehicles have a voltage regulator which cuts the high voltage down to somewhere between 14 and 15v, typically. The "raw" voltage of the alternator is typically around 17 or 18v. Sometimes, the regulator is external, and it's possible to tap the higher voltage directly. On other vehicles, the regulator is inside the alternator's casing and it's not easy to get to.
That said, even if you get access to a voltage high enough, LiPo batteries are notoriously sensitive to charge process. Most LiPo charging circuits switch between constant voltage and constant current at different points in the charging cycle, and involve active circuitry to monitor the charge and cut it off when the battery is full. It's dangerous to take a "dumb" approach of just applying a nominal voltage to a LiPo battery (as opposed to a car battery, which is quite happy to just get a constant 14.5v from the alternator, no matter what it's current state is). That's an important advantage of a traditional car battery, you can just directly connect it to the circuit with zero "overhead" in terms of charge or discharge management.
Further, it's fairly easy to damage LiPo batteries by leaving them charged much above or below their nominal rating and they have pretty limited lifetimes in terms of charge cycles. They also need to be balanced from cell to cell as they're cycled, which is impossible to do without a purpose-built charge controller. LiPos are also fairly sensitive to environmental conditions, and probably won't survive long in an automotive environment. Not to mention the significant fire hazard if or when they do fail.
When you say you're trying to make a "car battery" it's not super clear what your goal is for this battery, but if you are trying to replace a typical 12v battery with a 4s LiPo, that's probably not a good idea. Besides the above points, a LiPo's maximum current will be nowhere near close to what a purpose-built automotive battery is capable of - it may not be sufficient to turn a starter motor even under ideal conditions, much less power one on a really cold day or under other bad conditions.