Turns out that Page has its own .map() method, to which you can pass a method reference to do the conversion.
Here is how I ended up doing the conversion.
final Page<ContactDto> contactDtoPage = contactPage.map(this::convertToContactDto);
The convertToContactDto method simply creates and returns an instance of the class I’m trying to convert to:
private ContactDto convertToContactDto(final Contact contact) {
final ContactDto contactDto = new ContactDto();
//get values from contact entity and set them in contactDto
//e.g. contactDto.setContactId(contact.getContactId());
return contactDto;
}