David Montgomery and his girlfriend's offseason move to Detroit didn't get off to an auspicious start ... TMZ Sports has learned one of the couple's dogs was involved in a bloody biting incident just days after the new Lions running back got settled into his new Michigan home.
According to police documents, officers were called out to Montgomery's new Grosse Pointe Shores residence on June 4 ... after his and his GF's pitbull mix, Lola, was accused of escaping the house and attacking a smaller dog on the sidewalk.