How can I catch the Material Datepicker month pagination event?

Well, my another response is wrong, Let’s go to take another aproach.
Imagine we has a CustomTemplate Header. As we want that it’s looks like the original DatePicker header we will copy the template of this that is in gitHub of angular, but we will change the functions in button next an prev, so it’s like

<div class="mat-calendar-header">
  <div class="mat-calendar-controls">
    <button mat-button type="button" class="mat-calendar-period-button"
            (click)="currentPeriodClicked()" [attr.aria-label]="periodButtonLabel"
            cdkAriaLive="polite">
      {{periodButtonText}}
      <div class="mat-calendar-arrow"
           [class.mat-calendar-invert]="calendar.currentView != 'month'"></div>
    </button>

    <div class="mat-calendar-spacer"></div>

    <ng-content></ng-content>

                <!--see that we change previousClicked by customPrev-->
    <button mat-icon-button type="button" class="mat-calendar-previous-button"
            [disabled]="!previousEnabled()" (click)="customPrev()"
            [attr.aria-label]="prevButtonLabel">
    </button>

                <!--see that we change nextClicked by customNext-->
    <button mat-icon-button type="button" class="mat-calendar-next-button"
            [disabled]="!nextEnabled()" (click)="customNext()"
            [attr.aria-label]="nextButtonLabel">
    </button>
  </div>
</div>  

Now we defined our customHeader extends from MatCalendarHeader

export class ExampleHeader extends MatCalendarHeader<any> {

  /** Handles user clicks on the period label. */
  currentPeriodClicked(): void {
    this.calendar.currentView = this.calendar.currentView == 'month' ? 'multi-year' : 'month';
  }

  /** Handles user clicks on the previous button. */
  customPrev(): void {
    console.log(this.calendar.activeDate)
    this.previousClicked()
  }

  /** Handles user clicks on the next button. */
  customNext(): void {
    console.log(this.calendar.activeDate)
    this.nextClicked()
  }

Then , just only say that use this ExampleHeader

<mat-form-field>
  <mat-label>Custom calendar header</mat-label>
  <input matInput [matDatepicker]="picker">
  <mat-datepicker-toggle matSuffix [for]="picker"></mat-datepicker-toggle>
  <mat-datepicker #picker [calendarHeaderComponent]="exampleHeader"></mat-datepicker>
</mat-form-field>

See in stackblitz