He might be ill, and it might suit you to know it, but no contact with him, for this, could ever be straight enough. (18.104.22.168)
Kate Croy has spent a lifetime dealing with the deception of her awful father. The man might be ill or dying, but she can never be sure because her father seems to have built their entire relationship on lies, even when the lies don't matter. He's a super pathological liar.
"Ah, I shan't perhaps come up to your idea. It's only to deceive Susan Shepherd." (22.214.171.124)
When Milly asks Kate to do something bad with her, Kate expects something scandalous. But Milly just wants Kate to help her deceive Susan by hiding the fact that she visited a doctor. This might not be a big deal for Kate, but for Milly this lie is a sign of independence.
But his seeing it she didn't mind a scrap, and not a scrap either his knowing how she had left the dear lady [Susan] in the dark. She had come alone, putting her friend off with a fraud: giving a pretext of shops. (126.96.36.199)
Milly doesn't care if Sir Luke knows that she has deceived her Susan. Actually, Sir Luke appreciates Milly's efforts to be independent and to live life on her terms. It's part of his patented "live life to the fullest" cure.
She found after this, for the day or two, more amusement than she had ventured to count on in the fact, if it were not a mere fancy, of deceiving Susie. (188.8.131.52)
After deceiving Susan, Milly finds a sense of pleasure in realizing that she can do whatever she wants. She's spent the better part of the book so far under Susan's constant attention and observation, and is more than ready to spread her (dove) wings and fly on her own. It's just a shame that she only discovers this after she's learned about her illness.
"You can find out where they are. If I myself try," Mrs. Lowder explained, "I shall appear to treat them as if I supposed them deceiving me." (184.108.40.206)
Mrs. Lowder wants Milly to basically be her "spy" and to find out whether or not Merton has returned to London from New York. Maud suspects that Merton and Kate are meeting behind her back, but doesn't want to appear as if she's suspicious. She knows that if they find out she's conspiring against them, it'll end up bringing Kate and Merton even closer together.
But Milly's answer had prepared itself while Aunt Maud was on the stair; she had felt in a rush all the reasons that would make it the most dove-like; and she gave it, while she was about it, as earnest, as candid. "I don't think, dear lady, he's here." (220.127.116.11)
Once she knows about her illness, Milly starts becoming a lot more independent, and this independence is reflected largely in her willingness to deceive people. For example, she knows almost for a fact that Merton is back in London and meeting with Kate. But she decides to deceive Aunt Maud by telling her the opposite. We can tell that she resents Aunt Maud's attempt to make her into a puppet, and so she responds with deception.
"He's to be told, please, deceptively, that I'm at home, and, you, as my representative, when he comes up, are to see him instead." (18.104.22.168)
Milly has already deceived her friend Susan and the overbearing Maud Lowder, but she takes it up a notch when she asks Susan to lie for her to Sir Luke Strett. Rather than stay in her house while Susan hears the news about her health, Milly decides to take life into her own hands and visit an art gallery where no one will recognize her.
"It seems to me […] that it's in detail we deceive [Aunt Maud]." (22.214.171.124)
Merton knows that Aunt Maud is aware of all the time he spends with Kate. In this sense, he has been completely honest. But he also knows he has deceived Aunt Maud in not telling her that he and Kate are engaged. Merton isn't willing to tell outright lies, but he is willing to not tell the truth if he's not asked for it. He has what you'd call a passive attitude toward deception.
"Well, because [Sir Luke] doesn't regard it as a trick. He could understand your action. It's all right, you see." (126.96.36.199)
Sir Luke totally understands when he learns about Milly's newfound interest in deceiving people. He (more than anybody) sympathizes with her efforts to develop her independence before she dies.
"You know what I mean. We've told so many lies."
Oh, at this his head went up. "I, my dear, have told none!" (188.8.131.52-60)
Kate tries to incriminate Merton in all of the lies she has told, but Merton insists that he has never told anyone any lies, unlike the people around him. In fact, he's the only person in the text who seems completely unable to tell a direct lie. When it comes to making Milly happy on her deathbed, though, it's debatable whether Merton's strict honesty is goodness or selfishishness.