The governor, a Democrat, declined to identify the family member, but said during a video news conference it wasn't First Lady Tammy Murphy.
"Per guidance from public health officials, the contact tracing process has begun to notify everyone who may have come into contact with the Governor's family member during the potential infection window," a statement said.
It was unclear whether the family member lives with Murphy. The governor and first lady have four children, but Murphy declined to specify details.
He received a test Wednesday that came back negative.
His office said the person doesn't qualify as a close contact, but "out of an abundance of caution" he was voluntarily quarantining.
Murphy said the person who tested positive hadn't been to any large indoor gatherings, and that the positive test likely stemmed from a smaller gathering. He said his family takes precautions, like keeping distance and opening windows, even at home.
It's the second time the governor has gone into isolation. In October, he abruptly left an event, saying he just found out a person he was in contact with over the weekend tested positive for the virus. Murphy tested negative after contact with that person.
Video from October 2020:
"From the beginning, the Governor has taken every precaution to limit the spread of COVID-19. Today's exceedingly cautious steps are part of that ongoing commitment," the statement said.
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The Associated Press contributed to this report.