The most common quality of great people is generosity

[Barque]

No. "Great by being good" doesn't sound natural. I've said already that your original sounded all right with a slight change.

The longer you make it, the more likely it is to be unnaural.

 

[ahsanul.irfan]

It doesn't work.

You might stick to your original, after removing "the" from before "people".

"The most common quality of people who are great by doing good is generosity, and the most common quality of people who can't be great by doing good is intolerance."

I hope this one is OK.

 

[Barque]

"Great by being good" doesn't sound natural.

I hope this one is OK.

You've just changed "being" to "doing".

As I said earlier, you could just stick to your original with the change I suggested.

Or you could try
Great people are often generous, and people who are incapable of being great are often intolerant.

I'm not sure it's true, at least the second part. It also depends on what you mean by "great", as someone else has already pointed out.

 

[Tarheel]

What do you mean?

If I was going to say something like that that's how I would phrase it.