Node view mode in page template suggestions
The hook hook_theme_suggestions_HOOK is the one you should implement. 

When you have a node of type page (labeled as Basic Page from the default profile), then the HOOK is node and the specific hook you would implement is hook_theme_suggestions_node(array $variables)

As @pinueve points out, there is already a specific suggestion that includes the custom display mode: node--page--CUSTOM_VIEW_MODE.html.twig.

Also, as @pinueve suggests, the easy way to find the template suggestions is by enabling the debugging mode. I tested this using a custom view mode called customvm that I created, so I can see these suggestions in the browser by inspecting the source code:

* node--17--customvm.html.twig
* node--17.html.twig
* node--page--customvm.html.twig
* node--page.html.twig
* node--customvm.html.twig
x node.html.twig


Alter hook

Note that there is also an alter version of the hook [HOOK_theme_suggestions_node(array &amp;$suggestions, array $variables)][3], you should use this if you are trying to alter the existing suggestions, maybe changing the priorities.

Suggestions for a page from a custom controller

The page HOOK is meant to be used when the output comes from a custom controller and route, like the one in this example:
https://www.drupal.org/docs/8/api/routing-system/introductory-drupal-8-routes-and-controllers-example

In this case the HOOK is page and you would implement these hooks: 


HOOK_theme_suggestions_page(array $variables)
HOOK_theme_suggestions_page_alter(array &amp;$suggestions, array
$variables)