[Vuejs]-Is there a way to NOT refresh the Page after Updating the data? Laravel 8 and Vue

Please next time add all relevant code to let us know what are you trying to achieve.

First of all, please note that data provided from props should not be mutated because of an anti-pattern. Said that you have to create a deep copy within your component in order to change its content.

Assuming that you are working in just 1 component, where you have your table listing all employees, you can do something like this.

<template>
    <div>
        <table>
            <tr v-for="item in employeeList" :key="item.id">
                <td>name: {{ item.name }}</td>
                <td>address : {{ item.address  }}</td>
                <td>position : {{ item.position  }}</td>
                <td><button class="btn btn-warning" @click="edit(item)"><i class="fa fa-edit"></i></button></td>
            </tr>
        </table>

        <div class="modal fade" id="editModal" tabindex="-1" role="dialog" aria-labelledby="exampleModalLabel" aria-hidden="true">
            <div class="modal-dialog" role="document">
                <div class="modal-content">
                  <div class="modal-header">
                      <h5 class="modal-title" id="exampleModalLabel">Employee Edit</h5>
                  </div>
                  <div class="modal-body">
                      <div class="form-group">
                          <label>Name</label>
                          <input type="text" class="form-control" v-model="form.name">
                      </div>
                  </div>
                  <div class="modal-footer">
                      <button class="btn btn-success" @click="update()">Save</button>
                  </div>
                </div>
            </div>
        </div>
    </div>
</template>

<script>
export default {
    props: {
        employee: Array
    },

    data: () => ({
        employeeList: [],
        form: {}
    }),

    mounted () {
        // Since changing a props is anti-pattern, we use a local data which can be manipulated
        this.employeeList = [...this.employee]
    },

    methods: {
      edit(item){
          // Assign the clicked item to form data
          this.form = item
          $('#editModal').modal('show')
      },

      update(){
          axios.put(`/employees/${this.form.id}`, this.form)
            .then(function (response){
                alert('Employee Updated') 
                // Find the employee index in employeeList array
                const updatedEmployee = response.data
                const index = this.employeeList.findIndex(x => x.id === updatedEmployee.id)
                // If employee is found, then proceed to update the array object by using ES6 spread operator
                if (index !== -1) {
                    this.employeeList = [...this.employeeList.slice(0, index), { ...updatedEmployee}, ...this.employeeList.slice(index + 1)]
                }             
            })
            .catch(function (error){
                console.log(error)
            })
      }
    }
}
</script>

Code is self-explanatory, but just in case I will clarify a little bit:

1. Because we can not mutate the props employee, we copy the array to local data by using the ES6 spread operator in mounted() hook.
2. When you click the button to edit an employee, you assign the item to the form data. Now you have the form with all employee data to be shown/changed anywhere.
3. Once success API response, since you are updating, you look for the array object and replace the whole array to avoid reactivity issues. In case you are adding a new one, you just can push it by doing this.employeeList.push(updatedEmployee)

EDIT: please note that the above code is a suggestion about how to work with a clean code.
Anyway, right to your question, you can update your array in your axios response by doing

            .then(function (response){
                alert('Employee Updated') 
                // Find the employee index in employeeList array
                const updatedEmployee = response.data
                const index = this.employeeList.findIndex(x => x.id === updatedEmployee.id)
                // If employee is found, then proceed to update the array object by using ES6 spread operator
                if (index !== -1) {
                    this.employeeList = [...this.employeeList.slice(0, index), { ...updatedEmployee}, ...this.employeeList.slice(index + 1)]
                }             
            })
            .catch(function (error){
                console.log(error)
            })

during update remove

location.reload();

and add the below code

$('#editModal').modal('hide');

To display data follow the procedure, update data receive from response:

updateStudent(){
            axios.put('update_student',{
                id:this.id,
                name:this.editname,
                email:this.editemail,
                phone:this.editphone,
            })
             .then(response=>console.log(response));
             axios.get('all_students')
            .then(response => {
                this.data = response.data;
            });
        },

You can display the updated data like the below code:

<tr v-for="row in data"> 
  <th scope="row">1</th> 
  <td>{{ row.name }}</td> 
</tr>

Let’s create an item in data to assign the value we get from props. Next, let’s assign props data to the created element.
The page refresh issue will be resolved.