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It goes against the idea of NATO which is based off the United Nations Charter.

NATO is a defensive alliance where the respect for sovereignty and mutual defense is what makes it work.

Turkey going for annexation goes against the ideology of NATO by not respecting international sovereignty. NATO will never help them in a forceful annexation. But if everybody involved allows the annexation peacefully, eventually it will be accepted.

If Turkey annexes by force, there are 3 most likely possibilities: NATO kicks them out, NATO stays neutral and does not help (even when the Kurds/Syrians retaliate), or NATO keeps a defensive role at the border while ensuring humanitarian aid to the area.

Whether or not NATO recognizes the annexation will depend on a vote by the members (Most likely with a high approval requirement).

Article 6

For the purpose of Article 5, an armed attack on one or more of the Parties is deemed to include an armed attack:

on the territory of any of the Parties in Europe or North America, on the Algerian Departments of France 2, on the territory of Turkey or on the Islands under the jurisdiction of any of the Parties in the North Atlantic area north of the Tropic of Cancer;

on the forces, vessels, or aircraft of any of the Parties, when in or over these territories or any other area in Europe in which occupation forces of any of the Parties were stationed on the date when the Treaty entered into force or the Mediterranean Sea or the North Atlantic area north of the Tropic of Cancer.

I presume that the other NATO countries will refuse to recognize annexed areas as part of the territory of Turkey and so they will not be protected by the treaty. Note how the treaty already does not protect even internationally recognized parts of certain member countries (for example Hawaii).