Can we say
Conversation started at the shopping.
Conversation started while shopping.
Which is correct?
"Conversation started at the shopping" is grammatically incorrect. Shopping is a verb, so you can't be "at" it. Of course the word "shopping" is sometimes included in a noun, for example "shopping mall".
The "correct" phrasing would depend on the timing. Do you intend to show that two people began a conversation as they began shopping, or during their shopping?
Your alternative "conversation started while shopping" implies that the subject had already started shopping and then the conversation began.
If you want to show the conversation began right away, you could say either:
Conversation started when they began shopping.
Conversation started when they arrived at the shopping mall.
Alternatively, if you want to show that conversation began during shopping you could say either:
Conversation started while they were shopping.
Conversation started while they were at the shopping mall.