[Vuejs]-How can i able to handle two id's (1 jquery id and 1 vue js id)?

If change in ID will break your code why not use data attributes, its totally meant for situations like this.Modify your markup:

Add className to ul tag:

<ul id="flexiselDemo1" class="flexis1">

In your javascript replace $("#flexiselDemo1") with

$(".flexis1#flexiselDemo1")