The witness, who spoke on condition of anonymity out of fear for his safety, said he saw soldiers shooting after an explosion hit Maiduguri city Tuesday.
It remains unclear if others were shooting back.
Army spokesman Lt. Col. Sagir Musa said he was unaware of the shootings of the woman and child.
He said soldiers shot dead a man who was pushing a wheelbarrow that contained explosives. He added that nobody died in the blast.
Musa blamed the attack on a radical Boko Haram sect, which is held responsible for more than 660 deaths this year, according to an Associated Press count.
Two bomb blasts have been reported on Tuesday in Maidiguri, the Borno state capital.
According to eyewitnesses, one of the bombs, which was planted at the West End roundabout in Maiduguri has injured several members of the Joint Task Force.
Report says that the large bomb embedded at the West End roundabout, which was remotely controlled, exploded around 10 AM Nigerian time. A second blast was reported at Sabon Layi area in Gwange Ward. Details of the second blast are sketchy.
The JTF spokesperson, Lt. Col Musa Sagir, confirmed there were explosions in the city but said a full briefing will be made to the media as the JTF was still actively patrolling the city to determine those responsible for the latest attacks.
More details to come…