Spike yields to her when she grabs his arm, since even he recognizes that question as reckless. He doesn't know why he asked in the first place, only that he didn't know how to react to inadvertently revealing something she was better off not getting a hold of.

Joshua's reaction to someone leaving is pretty extreme, and he's sure that there's more to it than that, but he accepts it without pushing further. He can't, when Lucrezia keeps him so far on the defensive.

There's a possibility that she's talking about the other name she learned, and he's very careful with how he responds.

"Collete. I don't think I met her."