After opening the dialog, the 'open' event fires. You can call 'focus' method to focus on the textbox component.
@Component({
selector: 'app-root',
template: `
<eui-linkbutton (click)="dlg.open()">Open</eui-linkbutton>
<eui-dialog #dlg [title]="'dialog'" [closed]="true" (open)="onOpen()" [bodyStyle]="{padding:'20px'}">
<eui-textbox #tb></eui-textbox>
</eui-dialog>
`
})
export class AppComponent {
@ViewChild('tb') tb: TextBoxComponent;
onOpen(){
setTimeout(() => this.tb.focus())
}
}