As @Pierre Said already given the answer. Here is more details about how Vue works.
Vue uses this.$refs to interact with DOM reference, you can use this or use app.input1 to set vue variable, where app is Vue object and input1 in data element
var app = new Vue({
el:'#app',
data:{
input1: ''
},
methods:{
method1: function (){
console.log("Insidide Vue: ",this.$refs['input1'].value);
}
}
})
function method1(e){
app.input1 = $(e).val();
console.log("Outside Vue: ",$(e).val());
}
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/vue/2.5.17/vue.js"></script>
<div id="app">
<input @change="method1" onchange="method1(this)" ref="input1">
<div>input1 : {{input1}}</div>
</div>