The Republican governor said he would not comment on the president-elect's staff appointments, which have been criticized by senior Maryland Democrats as "deeply troubling."
He addressed what Maryland Attorney General Brian E. Frosh called an "upwelling" of hate crimes by acknowledging "some things are happening across the state."
But he also said the Maryland State Police have received no reports about such incidents.
He said the only act of violence he's heard about involved a Trump supporter getting beaten up. Last week, a 15-year-old wearing a "Make American Great Again" hat was attacked at a Rockville High School while students protested Trump's victory.
"We would not like to see any hate crimes on either side of this issue," Hogan said.
Hogan was one of two Republican governors to say publicly that he would not vote for Trump. He issued a statement after the election supporting Trump and urging unity, but Monday marked the first time he took questions from reporters.
State Treasurer Nancy K. Kopp, a Democrat, called on Hogan last week to continue his "open opposition" to the president-elect.
"Everyone ought to just take a deep breath and give the new administration a chance," Hogan said. "He's going to take office in a couple of months, and we're going to see what he's going to do. People have every right to protest peacefully, but we're not going to stand for people breaking the law."
The governor also addressed the controversy over a message that was retweeted by Baltimore County schools Superintendent Dallas Dance. The tweet called for teachers to comfort students who might feel alienated by Trump's rhetoric. It mentioned comforting "non-white" students, which sparked some Republican lawmakers to call for Dance's resignation.
"I don't think it was a very smart thing for him to do," Hogan said, adding that some people "misinterpreted" the superintendent's remark.
"I'm sure he probably regrets sending the tweet out. But it's really a lot to do about nothing," Hogan said. "I don't think anything's going to come of it."