I bought a car manufactured by Toyota.
How would you say "a car manufactured by Toyota"?
I bought Toyota's car.
I bought a car of Toyota.
I bought a car from Toyota.
I bought a Toyota.
or is there a better alternative?
As Janus Bahs Jacquet points out, it's common to call a product by the manufacturer's name when it's closely identified with the manufacturer.
Thus you can say "I bought a Toyota."
If you say "I bought a car manufactured by Toyota" you could mean that you bought a Lexus, which is also manufactured by Toyota.
If you say "I bought a car "from Toyota," you imply that Toyota is the seller of the car, which it may or may not be. You may have bought the car second-hand or from a dealer rather than directly from the manufacturer.
"Toyota's car" and "a car of Toyota" are both odd-sounding and not idiomatic. However, it's possible to say "I like Toyota's cars", meaning that you like the various models and brands of cars that Toyota produces. You might say "I like GM's cars--the Corvette and the Volt and even the SUV's."