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On my website, there is a side menu, and in the side menu, there are a few buttons that contain submenus. Within those buttons containing submenus, there is also a symbol '<'. When I click on one of those buttons, the submenu opens, and when I click on the word itself, I am redirected to that page.

I want the submenu to be displayed only when I click on the '<' symbol. How can I fix this in the Menu_rendered server event?"

See options of AdminLTE's Treeview plugin, the trigger option should be what you want. As the docs' tip says: you can use any option via the data-attributes.

In which server event can i do that?

The treeview is outputted by an Auto JS Template in the _Layout.cshtml, look for data-widget="treeview". Test your code by modifying _Layout.cshtml first. When your code works, you can replace the auto js template by Page_Foot server event.