[Vuejs]-Laravel vuejs form submit returns to the same page with query string

try this.$router.push()

https://router.vuejs.org/guide/essentials/named-routes.html

submitMobile: function() {
    this.$http.post('/customer/send-sms', this.formData).then(function(response) {
    
    this.$router.push({
        name: "route_name", // tis route name you need to add 
        query: { mobile: this.mobile },  
    });

    }, function() {
    console.log('failed');
    });
}