I have created forms in HTML (shown image below), and I want to integrate this form in FullCalendar. This form will show when I wanted to add new event. Can you give me some idea? If possible, can I do this without modifying the plugin?

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You could put the form in a lightbox and show it, when the user clicks on a day. Fullcalendar provides an event for this: http://arshaw.com/fullcalendar/docs/mouse/dayClick/
Edit: Or what exactly do you mean by integrate?