Alger responded to the message asking which image would be used, and how it would be printed.
Appleby explained that the image, which is of Alger in a black one-piece, would be printed on a poster.
The young model "quickly replied" with her concern, explaining why she wasn't comfortable with the brand using the particular image of her in a swimsuit.
"I just don't really like that photo of myself," Alger wrote.
After trying to bolster Alger's confidence by replying: "What! You look gorgeous - what don't you like about it..." Appleby continued: "Don't you like our swimwear?"
The message chain then shows a follow-up from Appleby: "Well don't have the decency to reply then. So disappointed Stevie - I personally chose you because you seemed like such a nice girl.
"This is such a rude and immature way to treat us.
"You don't seem to be self conscious in your latest post with the way you are portraying your self image - sitting with your legs so far apart," Appleby commented referring to the below image.
"It seems like you are just like a lot of other Insta girls - so fake. From Allison."
"My boyfriend and all my friends will be there competing so I'm just a little self conscious but I know I can't really do anything about it."
"Shocked" by her response, Mrs Appleby questioned what it was about the photo Ms Alger disliked.
"You look gorgeous - what don't you like about it. Don't you like our swimwear?", Appleby wrote.
After not receiving a response from Ms Alger, Mrs Appleby replied with what has been labelled a "slut-shaming" remark towards the young model.
Ms Alger then responded, saying the exchange was one of the "meanest and rudest messages" she'd ever received.
"I think it's fair for every girl to feel self conscious from time to time," she added.
On Wednesday, Ms Alger's boyfriend made the conversation public on his Instagram account, a move Mrs Appleby says is a blatant attempt at "ruining her brand".
"This is pretty cruel," she told news.com.au. "She shouldn't have taken a private conversation for her boyfriend to then publish on Instagram."
Mrs Appleby, who has owned the swimwear brand for 14 years, said she doesn't believe the young model is self-conscious in any way.
Speaking of various images on Ms Alger's Facebook page, the designer said she couldn't understand the model's problem with Aquadiva's photo.
"It seems strange to me that she's comfortable sitting there in white tights and looking very confident in a very un-lady like pose," she said.
"Yet we have her in a black one-piece looking stunning. I just think she no longer wants to be associated with our brand."
Mrs Appleby said while she has apologised to Ms Alger for "stupidly lashing out" several times on Instagram, she had no intention on 'slut-shaming' her.
"I am old enough to know better, and I should've known better," she said.
"I didn't even know there was such term as 'slut-shaming'. I didn't call her a name, I just made a statement about a photo on her Instagram."
The photographer who took the images of Ms Alger and didn't want to be named, said the whole ordeal was just a desperate attempt at getting more followers.
"This girl is not self conscious," he told news.com.au.
"I photographed her, she just doesn't want to be associated with the swimwear anymore.
"It's a terrible lie and she's done a horrible thing to this family."
The photographer said that on the day of the photoshoot, Ms Alger "loved" the photos and there was no trouble working together.
"She's so full of it," he said.
"She's gained 3000 followers overnight, and if she felt so passionate about it - why didn't she post the messages instead of her boyfriend?
"I think this is unprofessional, and I'd be surprised if anyone booked her again."
Ms Alger declined to comment any further on the matter.
