Edit in JSFiddle

<div id="app">
    <div id="calendar-nav">
        <i class="glyphicon glyphicon-menu-left" v-on="click: moveLastMonth"></i>
        <span>{{calData.year}} - {{getMonthName(calData.month)}}</span>
        <i class="glyphicon glyphicon-menu-right"  v-on="click: moveNextMonth"></i>
    </div>
    <table id="calendar" class="table table-bordered">
        <thead>
            <tr>
                <th v-repeat="week: weeks">{{week}}</th>
            </tr>
        </thead>
        <tbody>
            <tr v-repeat="week: calendar">
                <td v-repeat="day: week">{{day.day}}</td>
            </tr>
        </tbody>
    </table>
</div>

/* カレンダーナビのスタイル */
#calendar-nav {
    text-align: center;
}

#calendar-nav span {
    display: inline-block;
    width: 200px;
}

#calendar-nav i:hover {
    cursor: pointer;
}

/* カレンダーのスタイル */
.table th, td{
    text-align: center;
}

#calendar th:first-child {
    background-color: #FEEEFF;
}
#calendar td:first-child {
    background-color: #FEEEFF;
}
#calendar th:nth-child(7) {
    background-color: #DFFFFF
}
#calendar td:nth-child(7) {
    background-color: #DFFFFF
}

#calendar td:hover {
    opacity: 0.6;
}

new Vue({
    el: '#app',
    data: {
        weeks: ['Sun', 'Mon', 'Tue', 'Wed', 'Thu', 'Fri', 'Sat'],
        calData: {year: 0, month: 0}
    },
    created: function (){
        var date = new Date();
        this.calData.year = date.getFullYear();
        this.calData.month = date.getMonth() + 1;
    },
    methods: {
        getMonthName: function(month) {
            var monthName = ['January','February','March','April','May','June','July','August','September','October','November','December'];
            return monthName[month - 1];
        },
        moveLastMonth: function() {
            if (this.calData.month == 1) {
                this.calData.year--;
                this.calData.month = 12;
            }
            else {
                this.calData.month--;
            }
        },
        moveNextMonth: function() {
            if (this.calData.month == 12) {
                this.calData.year++;
                this.calData.month = 1;
            }
            else {
                this.calData.month++;
            }
        }
    },
    computed: {
        calendar: function () {
            // 1日の曜日
            var firstDay = new Date(this.calData.year, this.calData.month - 1, 1).getDay();
            // 晦日の日にち
            var lastDate = new Date(this.calData.year, this.calData.month, 0).getDate();
            // 日にちのカウント
            var dayIdx = 1;

            var calendar = [];
            for (var w = 0; w < 6; w++) {
                var week = [];

                // 空白行をなくすため
                if (lastDate < dayIdx) {break;}

                for (var d = 0; d < 7; d++) {
                    if (w == 0 && d < firstDay) {
                        week[d] = {day: ''};
                    } else if (w == 6 && lastDate < day) {
                        week[d] = {day: ''};
                        dayIdx++;
                    } else if (lastDate < dayIdx) {
                        week[d] = {day: ''};
                        dayIdx++;
                    } else {
                        week[d] = {day: dayIdx};
                        dayIdx++;
                    }
                }

                calendar.push(week);
            }
            return calendar;
        }
    }
});