here is a simple way to do it:

working plunker: https://plnkr.co/edit/uJ3n8XedvCCdeUHXKpwX?p=preview

first, add hammerjs to your module:

import { HammerGestureConfig, HAMMER_GESTURE_CONFIG } from '@angular/platform-browser';

provide it in providers

  providers: [{ 
                    provide: HAMMER_GESTURE_CONFIG, 
                    useClass: HammerGestureConfig 
                }]

sample module:

@NgModule({

  imports: [
    BrowserModule,
    BrowserAnimationsModule,
    FormsModule,
    ReactiveFormsModule,
    MaterialModule,
  ],

  declarations: [TabsOverviewExample],
  bootstrap: [TabsOverviewExample],
  providers: [{ 
                    provide: HAMMER_GESTURE_CONFIG, 
                    useClass: HammerGestureConfig 
                }]
})
export class PlunkerAppModule {}

then build your component like this:

import {Component, AfterViewInit, ViewChild, ViewChildren} from '@angular/core';
import { MdTabGroup, MdTab } from '@angular/material';


@Component({
  selector: 'tabs-overview-example',
  template:`
  <md-tab-group [selectedIndex]="selected"  (swipeleft)="swipe($event.type)" (swiperight)="swipe($event.type)">
    <md-tab label="Tab 1"><div class="content">Content 1</div></md-tab>
    <md-tab label="Tab 2"><div class="content">Content 2</div></md-tab>
    <md-tab label="Tab 3"><div class="content">Content 3</div></md-tab>
    <md-tab label="Tab 4"><div class="content">Content 4</div></md-tab>
    <md-tab label="Tab 5"><div class="content">Content 5</div></md-tab>
  </md-tab-group>`,
  styles:['.content{ height: 500px; background-color: yellow;}']
})
export class TabsOverviewExample implements AfterViewInit{
  @ViewChild(MdTabGroup) group;
  @ViewChildren(MdTab) tabs;
  tab_num = 0;
  selected = 0;
  SWIPE_ACTION = { LEFT: 'swipeleft', RIGHT: 'swiperight' };

  number_tabs
  ngAfterViewInit(){
    this.tab_num = this.tabs.length
    console.log(this.group)
  }
  swipe(eType){
    console.log(eType);
    if(eType === this.SWIPE_ACTION.LEFT && this.selected > 0){
      console.log("movin left")
      this.selected--;
    }
    else if(eType === this.SWIPE_ACTION.RIGHT && this.selected < this.tab_num){
      console.log("movin right")
      this.selected++;
    }
    console.log(this.selected)
  }


}

This worked for me

https://plnkr.co/edit/VC270mPwh2o8CbMt9u7k?p=preview

Template

<div class="md-content" flex md-scroll-y (swipeleft)="swipe(idx, $event.type)" (swiperight)="swipe(idx, $event.type)">
<md-tab-group md-stretch-tabs [(selectedIndex)]="selectedIndex" (selectedIndexChange)="selectChange()">
  <md-tab label="Tab 1" (swipeleft)="swipe(1, $event.type)" (swiperight)="swipe(1, $event.type)">
      Content 1
    </md-tab>
  <md-tab label="Tab 2" (swipeleft)="swipe(2, $event.type)" (swiperight)="swipe(2, $event.type)">Content 2</md-tab>
</md-tab-group>
</div>

Component

export class TabsOverviewExample {


  selectedIndex: number = 1;

  selectChange(): void{
    console.log("Selected INDEX: " + this.selectedIndex);
  }

  SWIPE_ACTION = { LEFT: 'swipeleft', RIGHT: 'swiperight' };

  // Action triggered when user swipes
  swipe(selectedIndex: number, action = this.SWIPE_ACTION.RIGHT) {
  console.log("swipe");
    console.log("number",selectedIndex);
    console.log("action",action);
    // Out of range
    if (this.selectedIndex < 0 || this.selectedIndex > 1 ) return;

    // Swipe left, next tab
    if (action === this.SWIPE_ACTION.LEFT) {
      const isLast = this.selectedIndex === 1;
      this.selectedIndex = isLast ? 0 : this.selectedIndex + 1;
      console.log("Swipe right - INDEX: " + this.selectedIndex);
    }

    // Swipe right, previous tab
    if (action === this.SWIPE_ACTION.RIGHT) {
      const isFirst = this.selectedIndex === 0;
      this.selectedIndex = isFirst ? 1 : this.selectedIndex - 1;
      console.log("Swipe left - INDEX: " + this.selectedIndex);
    }
  }
}

Here is my version of @ahmed-musallam's answer using this approach:

Template:

<mat-tab-group #tabGroup [selectedIndex]="selected" (touchstart)="swipe($event, 'start')" (touchend)="swipe($event, 'end')">
...
</mat-tab-group>

Component:

@ViewChild('tabGroup', { static: true }) tabGroup: MatTabGroup;
public selected: number;
private swipeCoord?: [number, number];
private swipeTime?: number;
swipe(e: TouchEvent, when: string): void {
    const coord: [number, number] = [e.changedTouches[0].clientX, e.changedTouches[0].clientY];
    const time = new Date().getTime();

    if (when === 'start') {
        this.swipeCoord = coord;
        this.swipeTime = time;
    } else if (when === 'end') {
        const direction = [coord[0] - this.swipeCoord[0], coord[1] - this.swipeCoord[1]];
        const duration = time - this.swipeTime;

        if (duration < 1000 //
            && Math.abs(direction[0]) > 30 // Long enough
            && Math.abs(direction[0]) > Math.abs(direction[1] * 3)) { // Horizontal enough
            const swipe = direction[0] < 0 ? 'next' : 'previous';
            switch (swipe) {
                case 'previous':
                    if (this.selected > 0) { this.selected--; }
                    break;
                case 'next':
                    if (this.selected < this.tabGroup._tabs.length - 1) { this.selected++; }
                    break;
            }
        }
    }
}

You need to switch 'previous' and 'next' for RTL direction.