Did any of the Kingsguard have children during their service?

Question

At least one member of the Kingsguard had children during his service;

Jaime Lannister.

But are there any more beside him?

Answer 1

There was Ser Lucamore Strong, Kingsguard to King Jaehaerys I Targaryen, who fathered a bunch of children and was sent to the Wall.

"...Gwayne Corbray, Alyn Connington, the Demon of Darry, aye. You will have heard of Lucamore Strong as well."

"Ser Lucamore the Lusty?" Ser Loras seemed amused. "Three wives and thirty children, was it? They cut his cock off. Shall I sing the song for you, my lord?"

_{A Feast for Crows, Jaime II}

Note that the Kingsguard's vows are modelled after those of the Night's Watch and that is why they do not father children.

It was Visenya, not Aegon, who decided the nature of the Kingsguard. Seven champions for the Lord of the Seven Kingdoms, who would all be knights. She modeled their vows upon those of the Night's Watch, so that they would forfeit all things save their duty to the king.

_{The World of Ice and Fire, The Targaryen Kings: Aegon I}

Bran was going to be a knight himself someday, one of the Kingsguard. Old Nan said they were the finest swords in all the realm. There were only seven of them, and they wore white armor and had no wives or children, but lived only to serve the king.

_{A Game of Thrones, Bran II}