A ketogenic diet is about having your body burn fat as its primary energy source, not sugar. This diet is often prescribed for weight loss, to improve energy, & mental clarity.
A gluten-free diet is about not eating any gluten, which comes from wheat, barley, & rye. This diet is often prescribed because there is evidence that the presence of gluten is harmful. The removal of gluten does often lead too … you guessed it: weight loss, improved energy, & mental clarity.
The ketogenic diet, because it’s low in carbs, naturally means that you’ll probably eat less wheat, barley, and rye (as those are carbs). But it is not required to remove those completely.
It is of course possible to be fully GF and keto at the same time.
More important than all of that is the quality of your food.
For example, if you’re going to eat wheat, then organic, sprouted wheat bread is a far better choice than standard white bread, which is not even food. The latter is also made from wheat, but it has zero nutritional value for your body, and probably has a lot of harmful crap added to it.
As you know I don’t like to prescribe diets at all, although I do advocate the removal of gluten when approporiate.
But no, the ketogenic diet is not the same as, nor does it mean that you must be, gluten-free.