We don't know Father Quinel for very long before he's struck down by John Aycliffe, but what we do know, we like:
Frail from his many years, Father Quinel had served in Stromford his entire life. Now he was small and wizened, with sparse gray tonsured hair. Some claimed he was the unwanted son of the previous Lord Furnival, who had provided him with the church living when Quinel was still a boy. (8.5)
Elderly Father Quinel is the only kindness Asta and her son knew during the thirteen years they lived in Stromford. Later, he doesn't believe Aycliffe's accusations against Crispin, which is perhaps part of the reason he's killed. Plus, like Crispin, Father Quinel may be an illegitimate son of the Furnival family, making him Lord Furnival's half-brother and Crispin's half-uncle. That would also make him a potential heir, which is perhaps the other part of the reason he's killed.
With regard to his function in the novel, Father Quinel is the one protector Asta's son has after his mother dies. The loss of Father Quinel leads Asta's son to believe that "God, I was certain, had completely abandoned me" (11.27). With Father Quinel out of the picture, Asta's son is truly friendless until he meets Bear.