Is it possible to change props value from method in Vue component?

What you are doing will throw a warning in Vue (in the console).

[Vue warn]: Avoid mutating a prop directly since the value will be overwritten whenever the parent component re-renders. Instead, use a data or computed property based on the prop's value. Prop being mutated: "propRoomSelected"

The value will actually change inside the component, but not outside the component. The value of a parent property cannot be changed inside a component, and, in fact, the updated value will be lost if the parent re-renders for any reason.

To update the parent property, what you should do is $emit the updated value and listen for the change in the parent.

Vue.component("navigation-form",{    template: '#navigation-form',    props: ['propRoomSelected'],    data: function () {      return {        roomSelected: this.propRoomSelected,      }  },  methods:{      updateCoachStatus: function(event){         this.$emit("update-room-selected", 67) ;      }  }})

And in your parent template listen for the event

<navigation-form :prop-room-selected='propRoomSelected'                 @update-room-selected="onUpdatePropRoomSelected"></navigation-form>

Here is an example.

This is a common pattern and Vue implemented a directive to make it slightly easier called v-model. Here is a component that supports v-model that will do the same thing.

Vue.component("navigation-form-two",{    template: '#navigation-form-two',    props: ['value'],    data: function () {      return {        roomSelected: this.value,      }  },  methods:{      updateCoachStatus: function(event){         this.$emit("input", 67) ;      }  }})

And in the parent template

<navigation-form-two v-model="secondRoomSelected">

Essentially, for your component to support v-model you should accept a value property and $emit the input event. The example linked above also shows that working.

Another approach is using a computed property for handling props:

{  template: '#navigation-form',  props: ['propRoomSelected'],  data () {    return {      roomSelected: this.computedRoomSelected,      changeableRoomSelected: undefined    }  },  computed: {    computedRoomSelected () {      if (this.changeableRoomSelected !== undefined) {        return this.changeableRoomSelected      }      return this.propRoomSelected    }  },  methods: {    updateCoachStatus (event) {      this.changeableRoomSelected = 67    }  }}