Hazel only sent their mother a text, providing her address without adding any explanation, but that address was quite detailed, thus it was easy to reach that place.
Stefan only wanted to come along since he had something to say to Hazel and Gaia thought that was not a big deal, since she assumed this would give another low blow to Hazel, showing her how pathetic she had become.
By and large, Gaia was positive that Hazel would end up in slums, surrounded by beggars and those lower class people. After all, she was literally kicked out of the family house. Gaia even generously brought a few things for her, some groceries and food to show how sincere she was.
The scenario in her head, of course, Hazel would be too strong willed to receive any help from her and this would show Stefan how bad she was for being so rude to someone, who just wanted to help her.