As
@PhilC said, Islam didn't exist during biblical times. The first infighting between the Abrahamic religions were the Jewish leaders persecuting Christians from the first century AD. This lasted up until the Judeo-Roman war of 67-70 AD that saw the destruction of the temple in Jerusalem. After that point, the remaining Jews who hadn't converted to Christianity had bigger issues to worry about - namely the very survival of their religion, since the line of Aaron had just been wiped out.
My knowledge of what was happening in the middle east from the founding of Islam to the crusades is fuzzy, but it can't have been all good since the crusades initially happened because of Muslims forcibly encroaching on Christian territory and waylaying Christian pilgrims. I assume Islam slowly spread and gained strength until the point when Muslim countries started eyeing up Christian lands of the Middle East and were able to make a serious play for them. All the crusades really managed to do in the Middle East was slow them down. If I remember correctly, the Christian Kingdom of Jerusalem barely lasted a century before being taken again by Muslims. Not all crusades were failures though - the Reconquista campaign that retook Iberia (modern day Spain and Portugal) for Christendom was a crusade.
Long story short, I don't think relations between the three have ever been that good, and while bad relations between Christians and Jews do go back to biblical times, those mostly seem to be resolved. Islam and Christianity have, at best, tolerated each other, but as Islam itself does not go back to biblical times, neither do their bad relations.
I'm also pretty sure "People of the book" was a way Muslims referred to Christians and does not refer to Jews or other Muslims, but I could be wrong about that.