Joseph, as usual, tells me that I’m being oversensitive. I think he’s got a crush on her, maybe..."
—Sebastian, writing about how Joseph thinks he's being oversensitive about Kidman
In the Consequence, it shows that when Kidman was first being transferred to the KCPD, Joseph wasn't to thrilled that Sebastian wanted to train her. He was shown to not want to have to deal with her. However, after that, he grows more respect for her and possibly even develops a crush on her.
This interesting due to the fact that in the memory that Juli sees, Sebastian is shown to really want her to come with them. But in the main game, it's shown that Sebastian thinks that dealing with her is babysitting duty and he starts to suspect's that she's hiding something. While in the memory Joseph is seen not wanting to deal with her and bring her with them. To accepting her in the main game and calling Sebastian oversensitive for saying that he thinks she's hiding something.
So it seems like they both switched their previous opinions of her. Sebastian going from giving her a chance, to becoming suspicious of her. And Joseph going from disliking the fact that Sebastian wants to take her with them, to respecting her and caring for her.
During the events of the games, it's shown in Chapter 5 of The Evil Within/ the ending of chapter one in The Assignment, when they find Kidman trapped in a chamber filling with water, Joseph like Sebastian is concerned about saving her from drowning. When the chamber opens and Kidman falls onto the floor, he's immediately there to help her, asking if she's alright. Showing that he cares about her well being.
There is a segment after this where Joseph becomes a haunted and Juli has to fight him. He's shown to be aggressive and angry in this form. She has to attack him three times before he turns back into a normal person. He then drops to the ground (possibly dead or unconscious). She then is shown to be upset about having to hurt him and says he should've just stayed out of the way.
After this Kidman is not seen physically interacting with Joseph until he and Sebastian find her leading Leslie through a park. (This scene is in both the Main Game and The Consequence.) She ends up accidentally shooting Joseph instead of Leslie when the earthquake happens, she doesn't stay behind to apologize for what she's done but rather she immediately runs after Leslie.
So it's been shown that they have a rather complicated relationship. They both have shown that they don't want to hurt each other. (At least when they're right in their minds). But Kidman shows that stopping Ruvik from exiting STEM is her top priority. Though it can be assumed that Kidman was sorry for accidentally shooting Joseph and that she would never purposely do that.