Well, I believe you can get more generals just by having more territory as it appears the game will give you more generals when your number of regions: number of generals ratio exceeds a certain amount. This will come in the form of adoptions or births. When I was blitzing my Greek campaign, towards the end, I was getting 1 adoption almost every turn.

But if you really only want more births, I suppose you can build fertility temples (ie Ceres) which tend to give any general in the city the Fertile set of traits, which increase the probability of getting offspring. There might be retinue members which do the same thing as well, but I'm not really sure.