Here is the quote: “Foiled of glory, [the Earl of] Nottingham took up the famous challenge of St. Ingelbert... offered to hold the lists against all comers in any form of combat for thirty days... He and his companions maintained the lists of St. Ingelbert with great courage...”
What is this challenge and what are its historical roots? Did it have anything to do with the saint himself? His Wikipedia page doesn’t have any mention of it.
Unfortunately it’s far less interesting than that…it’s a joust, in this case a famous one, that took place at Saint-Inglevert outside Calais. The name actually comes from something like “Sondingeveld”, the field owned by the family of a guy named Sondo. That meaning was obviously lost over time and the residents interpreted the name as if it referred to a saint, but there aren’t any saints named Ingelbert - or actually, there is one, Engelbert the archbishop of Cologne, who was murdered in the 13th century, but the name of the town is much older than that.
In 1390 there was a truce between England and France during the Hundred Years’ War, and some French knights, led by Jean Le Maingre the younger (also known as Boucicaut, like his father, John Le Maingre the elder) organized a jousting tournament for May 21. Some people thought this wasn’t a very wise idea since the truce wasn’t very old and the war was still fresh in everyone’s mind, but some English knights answered the challenge, including the Earl of Nottingham, as you mention, but also the Duke of Exeter, the half-brother of King Richard II. Richard himself did not come, but the king of France, Charles VI, was there - but in disguise, just as a spectator, not a participant.
Boucicaut was quite a famous knight. He visited the Holy Land and also fought in Spain and Prussia. He was named Marshal of France the next year in 1391, and in 1396 he participated in the Nicopolis crusade against the Ottomans, where he was taken prisoner.
Anyway, this was just a one-time tournament that happened to take place at Saint-Inglevert, and it was a famous tournament, but not an ongoing event and nothing to do with an actual saint.
I don't know of any sources about this in English but there is an article about in French: Élisabeth Gaucher, "Les joutes de Saint-Inglevert: perception et écriture d'un événement historique pendant la guerre de Cent Ans" in Le Moyen Âge 102 (1996).